Price (10-8), who came into the contest with a 1-7 record in 11 career starts against the Rangers, allowed two runs on seven hits in his fourth complete game of the season and eighth of his career.
The reigning AL Cy Young Award winner issued one walk and struck out four with a 118-pitch performance in the 10th "Game 163" in baseball history, which put the Rays into Wednesday's American League Wild Card Game at Cleveland.
Evan Longoria belted a two-run homer in third inning to give the Rays a 3-0 lead while Delmon Young and David DeJesus each added an RBI in the victory, as Tampa Bay finished the season with a 92-71 mark.
Texas starter Martin Perez (10-6) yielded three runs on four hits over 5 1/3 innings, allowing two walks while striking out five.
With the loss, Texas missed the post-season for the first time since 2010.