Jennings' hit boosts Rays over RangersRays rally to reclaim No. 1 wild card spot.
Texas has a half-game lead over Cleveland, which lost at Kansas City, for the second wild card.
Jose Lobaton opened the 12th with a single off Joseph Ortiz (2-2) and pinch-runner Sam Fuld went to second on Yunel Escobar's bunt. Fuld then scored on Jennings' hit to right center.
Brandon Gomes (2-1) struck out Ian Kinsler with runners on second and third with two outs in the top of the 12th for the Rays, who have won two of the first three games of the four-game series.
Texas took a 3-2 lead in the 11th when Elvis Andrus scored from first base on Adrian Beltre's two-out single.
The Rays tied on David DeJesus' RBI single in the bottom of the inning as Matt Joyce drew a two-out walk from Rangers closer Joe Nathan, pinch-runner Freddy Guzman stole second and scored on David DeJesus' single. It was Nathan's third blown save in 42 chances this season.
Sean Rodriguez, who had just two hits in his previous 16 at-bats against Rangers' starter Derek Holland, tied it at 2-all in the sixth with a two-run homer off the left-hander.
Rodriguez also made a diving catch in shallow left with one out in the ninth to take a hit away from A.J. Pierzynski.
Leonys Martin put Texas up 2-0 on a two-run triple off Chris Archer in the second. Jennings appeared to misread the drive in center, which went over his head.
Martin had two doubles and a homer in the Rangers' 7-1 win over Tampa Bay on Tuesday.
Archer gave up two runs and four hits over six innings. Holland, who went 0-3 with a 10.13 ERA in his previous three starts, allowed two runs and six hits in six innings.
NOTES: Tampa Bay LHP Matt Moore (15-3) will go against Texas RHP Yu Darvish (12-9) in Thursday night's series finale. ''This is the most fun time of the year for us,'' Moore said. ... Rays reliever Jesse Crain (strained right shoulder) is scheduled to pitch in an instructional league game on Thursday. ... Lance Berkman (hip and knee), 0 for 12 in five games since being reinstated from the DL on Sept 1, still can't swing right-handed but remains in the Rangers' left-handed DH mix. ''I know he hasn't done a heck of a lot, but he's one of those guys who can wake up on a cold day with snow out there and get a base hit,'' Texas manager Ron Washington said.