Mariners starter Hisashi Iwakuma shut out the Rays for eight innings, allowing four hits, before giving way to Rodney. David DeJesus led off the ninth with a home run to tie the game, and Rodney (1-2) gave up a go-ahead RBI single to Matt Joyce five batters later.
Price (4-3) retired the side in order in the bottom of the ninth to complete the victory for the Rays. He kept the Mariners off balance most of the night, allowing one run and six hits. Eight of his 12 strikeouts came on called third strikes.
Rodney had a 2-2 count on DeJesus when the designated hitter sent a fly ball down the right-field line, and all right fielder Stefen Romero could do was watch as the ball cleared the wall with a couple of rows to spare.
With two outs, James Loney golfed a single up the middle and was replaced by pinch-runner Sean Rodriguez. Desmond Jennings' hard grounder to short hopped past Brad Miller, and then Joyce's fly to right-center dropped in front of the diving James Jones to score Rodriguez.
Robinson Cano had a double, single and an RBI for Seattle.
It looked as though Price would come out on the wrong end of the pitchers' duel until the Rays' ninth-inning rally.
The Mariners took a 1-0 lead in the first inning when Jones led off with a double down the left-field line and later scored on Cano's groundout to first.
Iwakuma, making his third start after opening the season on the disabled list with a finger injury, gave up a double to DeJesus in the first, then retired 13 in a row before the Rays picked up a pair of singles in the fifth. Tampa Bay's only other hit off Iwakuma was Joyce's single to lead off the eighth, but Iwakuma finished off that inning and his night with a called strikeout of Wil Myers with the runner stranded on first.
It was the second straight eight-inning shutout outing for Iwakuma, who has now thrown 24 straight scoreless innings at home dating to last season.
NOTES: Before the game, Rays manager Joe Maddon said he was encouraging his players to use old-school cologne for the game in an effort to shake the team out of its slump. "Aroma matters," he said. Maddon added that he hoped the smells would conjure up warm memories of the players' fathers. "I'm hoping for this really awful odor in the dugout with all these scents tonight. That would be outstanding." ... Maddon also used the more traditional method of mixing up the lineup. Myers (leadoff), DeJesus (No. 2) and 2B Ben Zobrist (No. 4) were in those spots for the first time this season.