Ben Francisco's sacrifice fly in the seventh inning broke a 1-1 deadlock and Gimenez followed with his decisive two-run hit, helping the Rays to victory in the opener of a season-ending three-game series.
Ben Zobrist added a pair of hits, including a solo homer for Tampa Bay, which fell short of the playoffs after the Oakland Athletics clinched the second American League wild card spot later Monday with a win over Texas.
Alex Cobb (11-9) baffled the Orioles on two hits over seven innings, yielding only one run while fanning seven.
Fernando Rodney nailed down the win with his 47th save.
Baltimore's Wei-Yin Chen (12-11) took the loss, allowing four runs (one earned) on six hits in a 6 2/3-inning effort.
Celebrity Couples of 2014 [PHOTOS]