On July 20, 2012, Allen made his Major League debut for the Cleveland Indians after being called up from Columbus He pitched an inning of relief giving up two walks, no runs, no hits, and one strikeout. He struck out Baltimore Orioles third baseman Mark Reynolds for his first career strikeout.

In 2013, Allen went 6–1 with a 2.43 ERA and 88 strikeouts in 70.1 innings, averaging 11.3 strikeouts per 9 innings. His 77 games pitched was second in the American League, trailing only Joel Peralta’s 80 (and 2nd all-time for any Indians pitcher), and he was 2nd in the league in errors committed by a pitcher (3), and 8th in wild pitches (9). He tied for 6th in the 2013 BBWAA AL Rookie of the Year voting.

After multiple struggles by closer John Axford early in the 2014 season, Allen was moved into the closer role. Allen finished the season with 76 appearances (again 2nd in the American League; tied for 4th all-time for any Indians pitcher), a 6–4 record with a 2.07 ERA, 91 strikeouts (8th-most in a single-season by an Indians reliever), and 24 saves (9th) in 69 ⁄3 innings, averaging 11.8 strikeouts per 9 innings.

For the 2015 season, Allen pitched in 70 games (8th in the AL), leading the American League with 58 games finished, while recording 34 saves (6th), 99 strikeouts (2nd among AL relievers), and a 2.99 ERA in 69 ⁄3 innings, averaging 12.9 strikeouts per 9 innings.

Allen married Mallory Rundquist in November 2015. The couple have a son, who was born in 2017. His twin brother, Chad, played baseball for the University of West Florida.

In 2016, he had 32 saves (7th in the AL) in 35 save opportunities, going 3–5 with a 2.51 ERA. He struck out 87 batters (8th among AL relievers) in 68 innings, averaging 11.5 strikeouts per 9 innings.

In 2017, he had 30 saves (6th in the AL) in 34 save opportunities, finishing the season with a 3–7 record and a 2.94 ERA. He struck out 92 batters (9th-most by an AL reliever) in 67 ⁄3 innings, averaging 12.3 strikeouts per 9 innings.

With his 141st career save in July 2018, Allen broke Bob Wickman’s franchise record for saves. Allen lost his closer role to Brad Hand after struggling in the month of August. He ended the season with 27 saves (6th in the AL) in 32 save chances in 70 appearances, as he was 4-6 with a 4.70 ERA, with 80 strikeouts in 67 innings, averaging 10.7 strikeouts per 9 innings. He gave up the longest home runs of any pitcher in major league baseball in 2018, at 495 feet. Through 2018, his 149 career saves were 9th of all major league active pitchers, tops of all major league pitchers under 30 years of age, and tops of all Cleveland Indians pitchers ever. He became a free agent after the 2018 season.

On January 20, 2019, Allen signed a one-year, $8.5 million contract with the Los Angeles Angels. He began the season as the Angels closer, going 4/4 for the first couple of weeks of the season despite control issues. He was removed of the closer role on April 24 and later was placed on the injured list with a lumbar strain. He was designated for assignment on June 15 after compiling a 6.26 ERA 25 games, and was released on June 18.

On June 22, 2019, the Twins signed Allen to a $500,000 minor league contract for the remainder of the 2019 season. He was released on July 31, 2019.

On February 7, 2020, Allen signed a minor league deal with the Texas Rangers. Allen was released by the Rangers organization on July 24, 2020.

On July 31, 2020, Allen signed a minor league contract with the Chicago Cubs. He was released by the organization on September 3, 2020.