Graham played basketball for the Miami Hurricanes men’s basketball team under head coach Frank Haith from 2005 to 2009. Graham graduated from Miami in May 2009 with a double major in marketing and management, then stayed at Miami to take graduate classes while playing a season of football. In 2009, he played tight end and finished the season with 17 receptions for 213 yards and five touchdowns in appearances in 13 games.

According to scouts, Graham was “extremely athletic, with an outstanding combination of size and speed for the tight end position,” yet he was considered very raw since he only had one year of college football experience. Graham was drafted by the New Orleans Saints in the third round of the 2010 NFL Draft, with the 95th overall selection. He signed a four-year, $2.445 million rookie contract on July 28, 2010.

In his rookie season in 2010, Graham started in only five games, playing behind veteran tight end Jeremy Shockey. In a Week 9 game at Carolina, his first NFL game in his home state, Graham caught a 19-yard pass from quarterback Drew Brees to score his first career NFL touchdown, contributing to a 34–3 Saints rout. On December 19, against the Baltimore Ravens in Week 15, Graham recorded two receiving touchdowns for his first multi-touchdown game in the 30–24 loss. Graham finished the 2010 season with 31 catches for 356 receiving yards and five receiving touchdowns.

Graham started the 2011 season off strong with four receptions for 56 receiving yards and a receiving touchdown in a 42–34 loss to the Green Bay Packers on NBC Sunday Night Football. In Week 3, a 40–33 victory over the Houston Texans, he had four receptions for 100 receiving yards and a receiving touchdown for his first game reaching the century mark. The next week, he recorded 10 receptions for 132 receiving yards and a receiving touchdown in the 23–10 victory over the Jacksonville Jaguars. In the follow game, a 30–27 victory over the Carolina Panthers, he recorded eight receptions for 129 yards. He recorded his fourth straight game with at least 100 receiving yards in the next game against the Tampa Bay Buccaneers with seven receptions for 124 receiving yards in the 26–20 loss. He recorded six receptions for 54 receiving yards and two receiving touchdowns in the next game against the Indianapolis Colts, a 62–7 victory. Over the last nine games of the regular season, he totaled 54 receptions for 636 receiving yards and six receiving touchdowns. Overall, in the 2011 season, his first full year as a starter, Graham had 99 receptions for 1,310 receiving yards and 11 receiving touchdowns and made his first Pro Bowl. He became the first tight end in Saints history to have more than 1,000 receiving yards in a season. He set the franchise record for receptions in a season while tying Marques Colston in 2007 and Joe Horn in 2004 for the Saints franchise record for touchdowns in a season. During the Saints’ Week 17 win over the Carolina Panthers, Graham broke Kellen Winslow’s NFL record of 1,290 receiving yards in a season by a tight end. However, the record was broken later that day by New England Patriots tight end Rob Gronkowski, who finished the season with 1,327 yards. He was named to the Pro Bowl for the 2011 season. In addition, he was named a second-team All Pro by the Associated Press and Pro Football Focus.

The Saints finished with a 13–3 record and won the NFC South. In the Wild Card Round against the Detroit Lions, Graham finished with seven receptions for 55 yards and a touchdown. In the Divisional Round against the San Francisco 49ers, Graham had a fourth-quarter 66-yard touchdown to give the Saints the lead, but the Saints ultimately lost 36–32 on a last-second 49ers touchdown pass. Graham finished with five receptions for 103 yards and two touchdowns in the loss. After the season, Graham was ranked 14th overall by his peers in the 2012 NFL Top 100 annual player poll, which ranks the 100 best players in the NFL based on votes from NFL players and coaches.

Graham started the 2013 season with four receptions for 45 receiving yards and a receiving touchdown in a 23–17 victory over the Atlanta Falcons. In the next game, a 16–14 victory over the Tampa Bay Buccaneers, he had ten receptions for 179 receiving yards and a receiving touchdown. In the following game, a 31–7 victory over the Arizona Cardinals, he had nine receptions for 134 receiving yards and two receiving touchdowns. He earned NFC Offensive Player of the Week honors for his performance against the Cardinals. His hot streak continued in the next game against the Miami Dolphins when he had four receptions for 100 receiving yards and two receiving touchdowns in the 38–17 victory. For his great performances in the month of September, he earned NFC Offensive Player of the Month honors. He recorded his fourth consecutive game with over 100 receiving yards in the 26–18 victory over the Chicago Bears with ten receptions for 135 receiving yards. However, in the next game against the New England Patriots, he was targeted six times but did not record a reception in the loss. He was able to rebound in the next game against the Buffalo Bills with three receptions for 37 receiving yards and two receiving touchdowns in the 35–17 victory. In the following game, a 26–20 loss to the New York Jets, he had nine receptions for 116 receiving yards and two receiving touchdowns. During the second divisional matchup against the Atlanta Falcons, Graham became known for his trademark touchdown celebration where he would slam dunk the football over the goalpost, a reference to his basketball background. He infamously bent the goalpost celebrating his touchdown against the Falcons. Prior to the 2014 season, the NFL banned goalpost dunks and made it punishable as “unsportsmanlike conduct,” which results in a penalty and a fine. Graham became the first player penalized under the new rule and was fined $30,000 for dunking during the preseason game against the Tennessee Titans in August 2014. On December 8, against the Carolina Panthers, Graham recorded his fifth game with two receiving touchdowns on the season in the 31–13 victory. In the 2013 season, Graham recorded 1,215 receiving yards and led the NFL in receiving touchdowns with 16. He received numerous accolades for his successful 2013 season. He was named to the Pro Bowl, earned First Team All-Pro Honors, and was ranked tenth by his fellow players on the NFL Top 100 Players of 2014.

Under the terms of the NFL’s collective bargaining agreement, Graham became a free agent after the 2013 season. On February 28, 2014, it was reported that the Saints had placed a non-exclusive franchise tag on him, meaning that he would be allowed to negotiate and sign an offer sheet with another team; the Saints would then have the right to match that offer sheet and retain Graham, and if they did not, the other team would owe the Saints two first round draft picks. No such competing offer sheet was signed, but negotiations between Graham and the Saints were complicated by disagreement as to whether Graham should be treated for bargaining purposes as a tight end (his official position) or as a wide receiver (where he often lines up during games), since the difference in applicable compensation under the franchise tag is more than $5 million for the year. On July 2, 2014, an arbitrator ruled that Graham was indeed a tight end for franchise tag purposes.

Graham started the 2014 season with eight receptions for 82 receiving yards in the 37–34 overtime loss to the Atlanta Falcons in Week 1. In the next game, a 26–24 loss to the Cleveland Browns, he had 10 receptions for 118 receiving yards and two receiving touchdowns. On November 9, against the San Francisco 49ers, he had 10 receptions for 76 receiving yards and two receiving touchdowns in the 27–24 loss. Two weeks later, against the Baltimore Ravens, he had six receptions for 47 receiving yards and two receiving touchdowns in the 34–27 loss. In the 2014 season, Graham recorded 85 receptions for 889 receiving yards and ten receiving touchdowns. He was named to the Pro Bowl for the 2014 season and was ranked 31st by his fellow players on the NFL Top 100 Players of 2015.

On March 10, 2015, Graham was traded to the Seattle Seahawks for center Max Unger and the Seahawks’ first-round selection in the 2015 NFL Draft. Graham wore number #88 in Seattle as number 80 was retired in honor of Steve Largent. In his Seattle Seahawks debut, he had six receptions for 51 receiving yards and a receiving touchdown in the overtime loss to the St. Louis Rams. Two weeks later, in the 26–0 victory over the Chicago Bears, he had seven receptions for 83 receiving yards and a receiving touchdown. On October 18, against the Carolina Panthers, he had eight receptions for a season-high 140 receiving yards in the 27–23 loss. On November 29, 2015, Graham suffered a torn right patellar tendon. The next day on November 30, 2015, Graham was placed on injured reserve. Overall, he finished his first season with the Seattle Seahawks with 48 receptions for 605 receiving yards and two receiving touchdowns.

After a slow start to the 2016 season with four receptions for 53 receiving yards combined in his first two games, Graham had six receptions for 100 receiving yards in the 37–18 victory over the San Francisco 49ers. He followed that up with six receptions for 113 yards in the 27–17 victory over the New York Jets. On November 7, against the Buffalo Bills, he had eight receptions for 103 receiving yards and two receiving touchdowns in the 31–25 victory. Over the last eight games of the regular season, he had 27 receptions for 378 receiving yards and three receiving touchdowns. In 2016, Graham finished the season with 923 receiving yards, 65 receptions, and six receiving touchdowns. He was selected to his first Pro Bowl as a member of the Seahawks.

Graham started the 2017 season off with only nine receiving yards on four receptions combined in the first two games against the Green Bay Packers and San Francisco 49ers, but he recorded 11 receptions for 133 receiving yards in the two games after against the Tennessee Titans and Indianapolis Colts. In the next game against the Los Angeles Rams, he scored his first touchdown of the season as part of six receptions for 37 receiving yards in the 16–10 victory. In the next game against the New York Giants, he found the endzone again as part of a 51-yard performance in the 24–7 victory. In the following game, a 41–38 victory over the Houston Texans, he recorded two receiving touchdowns in the 41–38 victory. On November 9, against the Arizona Cardinals, he recorded two receiving touchdowns in the 22–16 victory. On December 19, 2017, Graham was named to his fifth Pro Bowl. Overall, he finished the 2017 season with 57 receptions for 520 receiving yards and ten receiving touchdowns. He was ranked 89th by his fellow players on the NFL Top 100 Players of 2018.

On March 16, 2018, Graham signed a three-year, $30 million contract with the Green Bay Packers.

In the 2018 season, Graham was in a position group with fellow tight ends Lance Kendricks, Marcedes Lewis, and Robert Tonyan. On September 9, 2018, on Sunday Night Football against the Chicago Bears, Graham was limited to two receptions for eight yards in his Packers debut. On September 30, 2018, in a game against the Buffalo Bills, Graham caught his first touchdown as a member of the Packers, a one-yard pass from quarterback Aaron Rodgers, which was also his 70th career touchdown reception. On October 15, 2018, against the San Francisco 49ers, Graham had five receptions for 104 yards, which was his first 100 receiving yard game since Week 8 of the 2016 NFL season. Graham caught his second touchdown of the season on November 4, 2018 against the New England Patriots. Overall, he finished the 2018 season with 55 receptions for 636 receiving yards and two touchdowns with 12 starts.

On March 12, 2020, Graham was released by the Packers.

Graham signed a two-year, $16 million contract with the Chicago Bears on March 26, 2020. Graham made his debut with the Bears in Week 1 against the Detroit Lions and caught three passes for 25 yards and a touchdown during the 27–23 win. In Week 3 against the Atlanta Falcons, he had six receptions for 60 receiving yards and two touchdowns in the 30–26 victory.