Lewis (1-0) surrendered one run, walked two and struck out six while lowering his post-season ERA to 1.67 in six starts for the Rangers, who took a 2-1 lead in their best-of-five ALDS series.
Neftali Feliz tossed 1 1/3 scoreless innings to close out the victory and gain his second save of the series.
The Rangers broke the contest open with four runs in the seventh inning. Mike Napoli socked a two-run homer and Josh Hamilton drove in two more with a single in the uprising, which gave Texas a 4-1 lead.
The Rays came back with a run in the bottom of the frame and Desmond Jennings' second solo homer of the game cut Tampa Bay's deficit to 4-3 in the eighth, but they could get no closer.
David Price (0-1) yielded three runs on seven hits over 6 2/3 frames for the Rays, who will try to avoid elimination in Tuesday's Game 4.