Morrow (9-6) was bidding for baseball's sixth no-hitter of the year and third against the Rays entering the ninth. With two out, Evan Longoria hit a grounder near second baseman Aaron Hill. Hill dove to his left and got a glove on it, but it got past him for a hit.
Toronto manager Cito Gaston left Morrow in the game, and the right-hander fanned Dan Johnson to end the game, his 17th strikeout of the game, a career high.
The victory was the fourth in four starts for Morrow, who issued a pair of walks in the 136-pitch effort.
Vernon Wells drove in the only run of the game with a first-inning single. He later dislocated a toe on his right foot while making leaping catch at the wall on a Ben Zobrist fly ball to end the sixth inning.
Andy Sonnenstine (2-1) took the loss after allowing only one run on three hits over 5 1/3 innings.
The defeat was a season-high fifth straight for the Rays, who dropped two full games behind the New York Yankees in the American League East.