The Rays trailed 4-3 when Ben Zobrist opened the ninth with a triple off the top of the right-field wall against rookie Danny Farquhar (0-1).
Matt Joyce singled to tie it, Evan Longoria doubled and Will Myers was intentionally walked to load the bases. Bourgeois singled to right over the Mariners' drawn-in outfield.
Joel Peralta (2-5) pitched one inning. Farquhar didn't retire any of the five batters he faced.
Myers' two-run homer, his ninth of the season, cut the margin to 4-3 in the sixth after a leadoff double by Longoria.
Rookies Brad Miller and Nick Franklin drove in three runs with back-to-back extra-base hits in the fifth to give the Mariners a three-run lead that held up until the ninth.
David Price failed in his bid to become the first Rays starting pitcher to win this month. Price pitched seven innings, giving up four runs on five hits while striking out seven. All four Mariner runs came on two-out hits.
Kendrys Morales drove in Seattle's first run with a single in the first. With two outs, Kyle Seager was hit by a pitch, stole second, moved to third on a wild pitch and scored on Morales' hit.
Facing the Mariners for the first time in his 138th major league start, Price then retired 12 straight before walking Dustin Ackley with two out in the fifth.
Humberto Quintero reached on an infield chopper, Miller tripled into the left-field gap and Franklin doubled.
Mariners starter Aaron Harang, pitching in his 33rd different major league ballpark, gave up only three singles until the sixth, when the Rays opened with four straight hits and finished him.
NOTES: RHP Alex Cobb will start for the Rays on Thursday, pitching in the majors for the first time since being struck on the right ear by a line drive exactly two months ago. ... Rays LHP Matt Moore threw 55 pitches in a bullpen session Wednesday, but the All-Star pitcher is not satisfied that he is 100 percent and will not come off the disabled list this weekend. Moore plans to throw again Sunday. ...Rays CF Desmond Jennings, who is on the disabled list with a fractured left middle finger, hit off a tee. He is eligible to come off the DL on Monday . . . Farquhar had recorded the first five saves of his career in his last five appearances for the Mariners. "He seems like he's mentally and physically ready and he's done a great job for us," acting manager Robby Thompson said before the game. "In most situations we're going to continue to give him the ball in the ninth inning." . . . The umpire crew of Wally Bell, Marvin Hudson, Marty Foster and Jordan Baker visited the All Children's Hospital as part of the Umps Care Charities Blue for Kids program.