The Rays lead Baltimore and Cleveland, who both lost, by two games for the second AL wild-card spot.
Trailing 1-0, Tampa Bay scored three times in the eighth. Rodriguez broke a tie with an RBI single and another run came home when center fielder Abraham Almonte bobbled the ball for an error.
The Rays began their 10-game road trip in sole possession of the first wild-card spot and a four-game lead on the second wild-card position. Tampa Bay completed the swing at just 3-7.
Evan Longoria was hit by a pitch from Yoervis Medina (4-5) to start the Rays eighth and Luke Scott walked. James Loney hit a tying double, and Rodriguez singled off Oliver Perez.
Rays starter Matt Moore got a no-decision despite tossing 6 1-3 innings of one-run ball. The left-hander allowed four hits, walked three and struck out three.
Moore hasn't lost since June 14, a span of nine decisions.
Jake McGee (4-3) got two outs and the win. Fernando Rodney closed for his 33rd save in 41 chances.
The Rays picked up an insurance run in the ninth on an RBI double by Desmond Jennings.
Mariners starter Erasmo Ramirez pitched into the seventh, blanking the Rays on four hits and working around four walks.
Kendrys Morales led off the Seattle second with his 20th homer of the season and 99th of his career.
The Rays went 4 for 10 with runners in scoring position, They hit .177 (17 for 96) in those spots during the road trip.
NOTES: Rays SS Yunel Escobar did not start for the first time since July 21. He pinch-hit in the eighth. Manager Joe Maddon said Sunday's pitching matchup gave him a good excuse to sit Escobar, who he wants to keep from completely wearing down by the end of the season. ... Seattle 2B Nick Franklin was not in the lineup on Sunday, a day after taking a knee to the head while attempting to field a fly ball. Mariners manager Eric Wedge said the decision was not entirely based on Saturday's collision, though, and Franklin is fine. Franklin struck out pinch-hitting in the ninth.