New York lost for just the third time in 12 games and saw its lead over Baltimore in the American League East reduced to one game. It was only the second shutout of the Yankees in the last four months.
Morrow (9-7) walked three and struck out three while throwing 97 pitches. Brad Lincoln and Darren Oliver completed the shutout.
Brett Lawrie put Toronto in front with a two-run homer in the third and J.P. Arencibia hit a solo shot in the eighth. Edwin Encarnacion drove in a run with a double in the fifth and another in the seventh on a ground out.
Ivan Nova (12-8) took the loss, giving up four runs on six hits in 4 2/3.
Both the Yankees and Orioles have six games remaining in the regular season. After playing three more games in Toronto, the Yankees will return home to finish the campaign with three against Boston.
Baltimore, which did not play Thursday, begins a three-game series at home against the Red Sox on Friday and then will travel to Tampa Bay for three more.