Price (4-1) came into the contest with an 0-6 record and a 5.30 career ERA in nine career starts against Texas, but he shook off those numbers Sunday while limiting the Rangers to two runs.
The left-hander yielded six hits, walked one and struck out six as the Rays took two out of three at Rangers Ballpark in Arlington.
Derek Holland (2-2) was touched for five runs on nine hits through seven frames for American League West-leading Texas, which dropped a series for the first time this season.
Josh Hamilton delivered an RBI single for the Rangers before leaving the game after the second inning with a stiff back.
2014: The Year in Music [PHOTOS]