New York completed the move for the southpaw on Thursday in exchange for infielder Brandon Drury and Minor League outfielder Billy McKinney. McKinney was the No. 20 prospect in the Yankees' system, according to MLB Pipeline.
Happ, 35, is 10-6 this season with a 4.18 ERA in 20 starts. The 12-year veteran began his career with the Philadelphia Phillies. He has also played for the Houston Astros, Seattle Mariners and Pittsburgh Pirates.
"We've always really liked Brandon, we talked a lot about him before he was a New York Yankee," Blue Jays general manager Ross Atkins told MLB.com. "It's the great thing about talent at this level, this stage of their careers, how much conviction you have that there will be a significant contribution."
"Brandon has withheld significant performance in the Major Leagues over several years and this year just hasn't had quite the opportunity and we feel like we have that moving forward."
Drury, 25, hit .176 with one home run and seven RBIs in 18 games this season for the Yankees. He also hit .292 with six home runs and 32 RBIs in 61 appearances this season in Triple-A.
McKinney, 23, was 1-for-4 in two games this season for the Yankees. He also hit .230 with 13 home runs and 35 RBIs in 62 games in the Minor Leagues.
Happ is due $13 million this season and is a free agent in 2019.