Curry was drafted by the Philadelphia Eagles in the second round (59th overall) of the 2012 NFL Draft. He was the highest selected player from Marshall since Darius Watts in 2004. Curry signed a four-year contract with the team on May 9, 2012. His first year was uneventful, he played 6 games, started zero, and finished with 9 combined tackles. He was given an increased role in his second year, finishing with 17 tackles, 5 assists, 4 sacks, and 2 pass deflections. Curry had a breakout season in 2014, where he played all 16 games. Although he did not start a game, and he only recorded 17 tackles and 2 assists, he recorded 9 sacks, second on the team only to Connor Barwin and led the team in forced fumbles, with 4. In a disappointing 2015 season, Curry was moved to outside linebacker and struggled to find a groove, totaling 12 tackles and 3.5 sacks in 16 games and no starts.

Curry had his breakout season in 2010, as a junior where he had a career high 12 sacks. As a senior in 2011, Curry had 74 tackles, 11 sacks, and 6 forced fumbles. The 11 sacks were the sixth most in the nation and the 6 forced fumbles were second. Curry became the first Marshall player to win C-USA Defensive Player of the Year since Albert McClellan in 2006.

Although there was speculation that Curry could leave the Eagles in free agency, the arrival of new defensive coordinator Jim Schwartz and the transition back into a 4-3 defense cemented his return to Philadelphia. On February 2, 2016, Curry signed a five-year, $47.5 million contract extension with the Eagles with $23 million guaranteed. The Eagles won Super Bowl LII against the New England Patriots 41-33. Curry recorded four tackles in the victory.

On March 16, 2018, two years into his five-year contract, Curry was released by the Eagles after declining to take a pay cut.

On March 19, 2018, Curry signed a three-year, $23 million contract with the Tampa Bay Buccaneers.

On February 12, 2019, after one season, Curry was released by the Buccaneers.

On March 21, 2019, after one season away from the Eagles, Curry was re-signed on a one-year deal.

On August 10, 2020, Curry re-signed with the Eagles on a one-year deal. He was placed on injured reserve on September 15, 2020. He was designated to return from injured reserve on October 7, and began practicing with the team again. He was activated on October 17.