Jimenez (4-3) notched his first win since May 11, throwing 72 of his 108 pitches for strikes. The right-hander scattered four hits, issued one walk and struck out seven.
Giambi bashed a two-run homer in the second inning and added a run-scoring single in the next frame.
Asdrubal Cabrera also provided a two-run shot as the Indians evened their three-game series.
Rays starter Chris Archer (0-1) was touched for five runs on seven hits over four full innings after being recalled from Class AAA Durham before the game.
Celebrity Families of 2014 [PHOTOS]