Damon connected with two out in the eighth, giving the Rays a four-run lead. Damon's third hit of the game came after shortstop Mike McCoy mishandled Evan Longoria's grounder for an error and Ben Zobrist flied out for the second out in the inning.
Rookie reliever Alex Torres (1-1) threw five shutout innings to earn his first big league win in his fourth game this season. The left-hander replaced Jeff Niemann to start the second and allowed three hits.
Damon also had a run-scoring double in the first as the Rays won for the second time in six games. Boston lost 9-1 to the New York Yankees earlier Saturday.
Jose Bautista hit his major league-leading 43rd homer for the Blue Jays, who had won seven of 11. Ricky Romero (15-11) gave up six runs — one earned — and six hits over 7 2/3 innings.
After Brandon Gomes and Joel Peralta worked scoreless innings, closer Kyle Farnsworth pitched a perfect ninth in a non-save situation. Farnsworth had been out since Sept. 11 due to a sore right elbow.
Tampa Bay went ahead 3-2 in the fourth on Zobrist's solo homer. He has gone deep three times over his last seven games after going 39 games without a home run.
Bautista put the Blue Jays up 2-0 during the first when he lined a 3-2 pitch from Niemann into the left-field seats. Toronto failed to extend the lead later in the inning when David Cooper bounced out with runners on second and third.
Niemann had his scheduled start pushed back from Thursday to Saturday because of a sore upper back.
A series of defensive blunders helped the Rays tie it at two in the bottom of the first. B.J. Upton singled one pitch after Cooper was charged with an error at first for misplaying a foul ball. Upton then scored when McCoy was given an error after a making an errant throw to first on Zobrist's two-out grounder.
Zobrist came home when Damon flared a ball to short center that dropped between second baseman Kelly Johnson and center fielder Colby Rasmus for a double.
Notes:
Toronto INF Brett Lawrie (broken right middle finger) will have his offseason program established a week after the season ends when he undergoes a follow-up examination. … Niemann gave up two hits and walked two during his 38-pitch inning. … Jays 1B Adam Lind missed his second straight game after returning home for the birth of his child. … Tampa Bay RHP Wade Davis (10-10) and Toronto LHP Brett Cecil (4-10) are scheduled to pitch in Sunday's series finale.