With the Rays trailing 4-3, Jose Molina led off the fourth inning with a single off Ivan Nova (9-3) and Rodriguez followed by blasting a 2-1 pitch deep into the left field bleachers, putting the Rays ahead to stay.
Jeff Keppinger added a two-run single for Tampa Bay, which defeated the Yankees at home for the ninth straight time.
James Shields (8-5) earned the win despite allowing four runs on 10 hits over seven innings.
Nova ended up allowing six runs (three earned) six hits over six full frames.
Dewayne Wise went 2-for-3 with a solo homer for the American League East-leading Yankees.