David Price went seven innings to get the win, ending the Rays' three-game losing streak.
New York began the night tied for the lead in the American League East with Baltimore. Tampa Bay climbed within three games of the AL's second wild card berth.
Price (18-5) gave up two runs on five hits with six strikeouts. He was making his first start in 12 days.
Tampa Bay trailed 1-0 heading into the fifth, but Elliott Johnson drove in the tying run with a single and Jennings followed with his RBI base hit to give the Rays the lead. The third run of the inning scored on a double play grounder off the bat of Evan Longoria.
The Rays padded their lead to the extent that they were able to survive a solo homer by Curtis Granderson in the fifth and a two-run shot by Alex Rodriguez in the eighth that brought the Yankees within 5-4.
Tampa Bay scored an unearned run in the ninth on a fielding error by shortstop Eduardo Nunez.
Fernando Rodney recorded the final five outs for the Rays to collect his 43rd save.
CC Sabathia (13-6) allowed four runs on six hits over 6 2/3 innings to lose his third straight decision.
Rodriguez went 2-for-5 and moved past Willie Mays into sole possession of 10th place on the all-time major league hits list.
Notable deaths of 2014 [PHOTOS]