Rodriguez's pinch-hit blow off Oliver Perez snapped a 1-1 tie and helped the Rays notch only their third win in 10 games.
The triumph moved them two games ahead of Cleveland and Baltimore in the race for the final American League wild card slot.
James Loney went 3-for-4 with an RBI double against Yoervis Medina (4-5) to start Tampa Bay's late three-run rally.
Jake McGee (4-3) earned the win after getting two outs in the seventh and Fernando Rodney wrapped up the triumph with his 33rd save.
Kendrys Morales homered for Seattle.