The Rays appeared on the verge of a fifth straight defeat in the finale of this three-game set after Oakland scored once in the top of the 11th, but Desmond Jennings came through with a two-out RBI single in front of Rodriguez's walk-off blast off Dan Otero (4-1).
Yoenis Cespedes plated Josh Donaldson with a sacrifice fly in the A's half of the 11th, two innings after he forced extras with an RBI double off former Oakland closer Grant Balfour in the ninth.
The game's earlier stages featured a terrific pitcher's duel between A's hurler Sonny Gray and Tampa Bay's Alex Cobb, who returned from an oblique strain to limit Oakland to three hits and a pair of walks over 6 2/3 brilliant innings in his first start since April 12.
Gray allowed just one run on five hits and two walks in eight strong innings of work.
Jennings finished 2-for-5 with two runs scored for the Rays. Donaldson and John Jaso each had two hits in the loss, just Oakland's second in its last 13 games.