In a tense contest loaded with playoff implications, Texas was unable to hold onto a one-run lead in the 11th and also put runners on second and third with one out in the 12th without scoring.
Tampa Bay then took advantage by getting to Joe Ortiz (2-2). Lobaton opened the Rays' 12th with a single, moved to second on a sacrifice by Yunel Escobar and scored when Jennings sent a 1-1 pitch to right field.
At the end of the evening, Tampa Bay led Texas by one game for the first wild card berth in the American League. The Rangers were in the second wild card spot, but had a lead of only a half game over Cleveland, one game over Baltimore and 1 1/2 games over both Kansas City and New York.
Texas took a 2-0 lead into the sixth, but the Rays tied it in that inning on a two-run homer by Sean Rodriguez.
Elvis Andrus scored all the way from first on a single by Adrian Beltre to put Texas in front in the 11th, but Tampa Bay came back to tie it in the bottom of the inning against closer Joe Nathan. Twice the Rangers were within a strike of winning the game, but Nathan gave up a two-out walk to Matt Joyce.
Joyce then stole second and scored on a single by David DeJesus to tie the contest.
The win went to Brandon Gomes (2-1), who struck out the only batter he faced in the 12th.