Sánchez signed with the New York Yankees as an international free agent in July 2009, receiving a $3 million signing bonus. Before the 2010 season, Baseball America ranked him as the Yankees’ seventh-best prospect. He began his professional career playing for the Gulf Coast Yankees. On August 18, 2010, he was promoted to the Staten Island Yankees of the Class A-Short Season New York–Penn League. He finished the season hitting .329/.393/.543 with eight home runs and 43 runs batted in (RBIs) in 47 games.

Before the 2011 season, Sánchez was ranked among the top prospects in baseball. He was rated as the second best prospect in the Yankees organization and 30th best prospect overall by Baseball America. He spent the season playing for the Charleston RiverDogs of the Class A South Atlantic League. He played in 82 games before suffering a season ending finger injury. He finished the season hitting .256/.335/.485 with 17 home runs and 52 RBIs.

Before the 2012 season, Sánchez was ranked as the Yankees fourth best prospect by Baseball America. He began the 2012 season with Charleston and received a midseason promotion to the Tampa Yankees of the Class A-Advanced Florida State League.

Sánchez competed with Austin Romine to serve as the backup catcher to Brian McCann on the Yankees’ Opening Day roster in 2016, but he began the season with Scranton/Wilkes-Barre. He was called up to the majors for one game in May, where he went 0-for-4. The Yankees again promoted Sánchez to the major leagues on August 3, and he recorded his first major league hit, a single off Hansel Robles, that same day. On August 10, Sánchez went 4-for-5, with his first major league home run and 3 singles. On August 16, Sánchez had his first two home run game against the Toronto Blue Jays. On August 22, Sánchez was named AL Player of the Week, after hitting four home runs with a .523 average. Sánchez became the first player in MLB history with at least 11 home runs and 31 hits in his first 23 career games. On August 29, Sánchez was named AL Player of the Week for the second straight week, and on September 3 MLB named him AL Player of the Month and Rookie of the Month. On September 21, Sánchez hit his 18th and 19th home runs in his 45th game, making him the fastest player to reach 19 home runs in the modern era.

For the 2016 season, Sánchez played in 53 games for the Yankees with a .299 batting average, 20 home runs, and 42 RBIs. On defense, in 2016 he had the best average pop time to second base (1.90 seconds) of all major league catchers. He finished second in the voting for AL Rookie of the Year, behind Michael Fulmer.

On April 8, 2017, Sánchez left the game after straining his right brachialis muscle in the bicep and was placed on the 10-day disabled list. He returned to the Yankees on May 5 after missing 21 games. On June 10, Sánchez hit the hardest home run of his career, a 115.1-mph laser to left-center field against Mike Wright of the Baltimore Orioles. The following day, Sánchez hit the farthest home run of his career, a 450-foot shot off Kevin Gausman of the Orioles. Sánchez was named to the 2017 American League All-Star team, his first, as a reserve. He participated in the 2017 MLB Home Run Derby, upsetting Miami Marlins outfielder Giancarlo Stanton 17–16 in the first round before losing to Minnesota Twins third baseman Miguel Sanó 11–10 in the second round. Prior to the competition, Sánchez engaged in a feud with Tampa Bay Rays first baseman Logan Morrison, who stated that Sánchez did not deserve an invitation to the Derby, having hit only 13 home runs so far compared to Morrison’s 24. Sánchez responded by saying “It’s not my fault he didn’t get selected.”

He finished the season batting .278/.345/.531 with 33 home runs, 90 RBIs, 20 doubles, 131 hits, 40 walks and 79 runs scored, and leading the majors in pull percentage (51.6%). On defense, in 2017 he had the best arm strength (87.8) and the third-best average pop time to second base (1.93 seconds) of all major league catchers, but tied for the major league lead in passed balls (16). In the postseason, he batted .208/.218/.415 with three home runs and eight RBIs as the Yankees lost in Game 7 of the ALCS to the Astros. At season’s end, Sánchez received the Silver Slugger Award after the season as the best hitting catcher in the American League.

On June 24, 2018, he injured his groin following an awkward landing at first base. He was placed on the disabled list, allowing backup catcher Austin Romine to fill his place and become the primary catcher. On July 23, after a game against the Tampa Bay Rays, Sanchez was criticized for lack of effort twice during the game. In the bottom of the first inning, a low pitch from Luis Severino bounced off Sanchez’s foot and rolled up the third base line. Sanchez jogged slowly to the ball, and Rays baserunner Jake Bauers took advantage of the lack of effort and scored after an errant throw from Sanchez. In the ninth inning, the Yankees were one run down with a chance to tie the game. Aaron Judge was at third, with Giancarlo Stanton at second, Aaron Hicks at first, and Sánchez up at bat. Sanchez hit a sharp ground ball to Daniel Robertson, who flipped it to Willy Adames, but Hicks hustled enough to be safe at second. Sánchez jogged slowly all the way to first, and Adames noticed, throwing to Bauers to get Sánchez out and end the game. Judge would have scored and tied the game but instead the Yankees lost to the Rays 6-7, due to Sánchez’s lack of hustle. On July 24, Sánchez went on the disabled list again due to re-aggravating the same groin.

On April 7, 2019, Sánchez hit 3 home runs in one game against the Baltimore Orioles. On April 27, 2019, Sánchez hit his first career grand slam against the San Francisco Giants.

On May 19, 2019, Sánchez hit his first career triple against the Tampa Bay Rays.

On August 23 2019, Sánchez hit his 100th career homerun against the Los Angeles Dodgers making him the fastest catcher to hit 100 homeruns and second fastest player of all time to hit 100 homeruns.

On September 3, 2019, Sánchez broke his own franchise record by hitting two home runs (number 33 and 34), a winning effort against the Texas Rangers. He now has the two highest single-season home run totals ever for a Yankees catcher, as well as becoming the second with multiple 30-homer years, joining Yogi Berra.

In 2019, Sanchez dropped his passed balls from the league lead of 18 in 2018 to 7. However on defense he led all major league catchers in errors committed, with 15.

Sanchez struggled during the 2020 season. In 49 games, he batted .147 with 10 home runs and 24 RBIs. By September, he was benched by manager Aaron Boone and relegated to the backup role. On defense, in 2020 he led all major league catchers in errors, with six.