The Athletics won the game in the bottom of the 19th on a walk-off Brandon Moss home run. Blevins also batted in that game, striking out against Barry Enright in the 18th inning. In 67 appearances in 2013, Blevins went 5-0 with 4 holds and a 3.15 ERA, striking out 52 in 60 innings. On December 11, 2013, Blevins was traded to the Washington Nationals for minor league outfielder Billy Burns. Blevins finished his 2014 season 2-3, with a 4.87 ERA, 66 strikeouts, and 23 walks, giving up 31 runs in 64 games in 57.1 innings pitched. On March 30, 2015, Blevins was traded to the New York Mets for outfielder Matt den Dekker. On April 19, 2015, Blevins was hit by a comebacker and suffered a distal radius fracture of the left arm and was placed on the 15-day disabled list. Blevins was recovering when he slipped on a curb and re-fractured his arm again, and missed the rest of the season. Blevins finished the 2015 season 1-0 with 4 strikeouts, no walks, no hits, and no hit batsmen in 5 innings pitched. , no player has pitched more innings in a season without giving up a hit, walk, or HBP. On December 15, 2015, Blevins signed a one-year, $4 million deal with incentives. In 2016 he was 4-2 with two saves and a 2.79 ERA and 52 strikeouts in 42 innings over 73 games. On February 9, 2017, Blevins signed a one-year contract with the Mets that included a team option for the 2018 season. In 2017 he was 6-0 with one save and a 2.94 ERA and 69 strikeouts over 49 innings in 75 games (3rd in the NL). In 2017 he threw a curveball 50.1% of the time, tops in MLB. On June 24, 2018, at Citi Field, after a late injury to Jason Vargas, Blevins made his first Major League start. He allowed home runs to the first two batters he faced, Enrique Hernández and Max Muncy of the Los Angeles Dodgers. Blevins became only the second pitcher since 1900 to allow home runs to both of the first two hitters in his first career start. He was pulled after two innings. On August 16 during the Mets' 24-4 rout of the Phillies, Blevins had his first MLB hit and RBI.