Jennings homered on the first pitch in the bottom of the 10th inning and the Rays beat the AL West-leading Texas Rangers 5-4 Wednesday.
The Rays started the day eight games behind Boston in the AL wild-card race. The Red Sox open a three-game series Friday night at Tampa Bay.
"It was big," Rays pitcher David Price said. "We can't really fall anymore behind. We've got to put some sweeps together."
"We can still make it really interesting if we put some pressure on them by winning those games," Davis said.
Jennings won this game with his ninth homer of the season, sending a pitch from Mark Lowe (2-3) into the left-field stands. The rookie had just four hits in his previous 38 at-bats.
"That's probably the most exciting feeling I've had in a while," Jennings said. "It was fun. I've been a little frustrated with myself, but pitchers have been making good pitches, too. That made me feel a lot better."
Texas tied it at 4 in the ninth when Ian Kinsler hit his second homer of the afternoon. Kinsler's 28th home run came with one out against Tampa Bay closer Kyle Farnsworth. Kinsler has seven homers in his last seven games.
"We had an opportunity," Rangers manager Ron Washington said. "Desmond Jennings ended it. The game was decided by one pitch."
The win was the 1,000th overall for the Tampa Bay franchise in the regular season. The Rays have lost 1,245.
Jake McGee (2-1) allowed a single during a scoreless 10th for the victory.
Ben Zobrist put the Rays ahead 2-0 with a two-run double in the first. He joined Milwaukee's Ryan Braun as the only big leaguers to have 100 doubles, 50 stolen bases and 50 homers over the last three seasons.
After David Murphy had an RBI grounder in the fourth, Kinsler made it 3-2 with a solo homer in the fifth off Price. Murphy had four RBIs in the Rangers' 8-0 win over Tampa Bay on Tuesday.
"That's a postseason lineup, one through nine," Price said.
"I knew we were going to come back," Holland said. "I had to keep my head straight. Overall, I thought I did a fairly good job of keeping us in the ballgame."
Mike Napoli cut the deficit to 4-3 with a solo homer in the Texas eighth.
Price joined teammate James Shields in reaching 200 strikeouts this season when he fanned Napoli to end the fifth. ... Texas LF Josh Hamilton, who tweaked his back Tuesday night, went 1 for 5 and struck out three times. ... Tampa Bay DH Johnny Damon (sore right foot) was out the lineup for the third consecutive game, but hit a fly ball as a pinch-hitter during the eighth. ... Rangers 3B Adrian Beltre extended his hitting streak to 13 games with a fourth-inning single. ... Rays C Jose Lobaton (sore left knee) could play this weekend. ... Bill McCrary, who played for Kansas City in the Negro Leagues from 1946-47, threw the ceremonial first pitch. ... Both teams are off Thursday.