The Rays lost for the sixth time in seven games.
A lack of clutch hitting plagued Tampa Bay throughout the game and the Rays had two runners thrown out at home.
One of those outs ended the game when Moises Sierra's throw from right field cut down Elliot Johnson at the plate.
The only run off Morrow (8-5) came on a homer by Desmond Jennings that gave Tampa Bay the lead in the third.
Toronto tied it in the bottom half of the third on a solo shot by Sierra and went in front in the fourth when Edwin Encarnacion hit his 35th homer of the season. He has the third largest total in the American League.
Casey Janssen picked up his 19th save thanks to Sierra's game-saving assist.
Jeremy Hellickson (8-10) took the loss, giving up the two home runs and two other hits in six innings.