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"It's one of those days where I couldn't really figure out why I was missing," Inge said. "I felt like I was taking good swings all day. As long as you can remember that that very last at-bat is going to be one that can make a huge difference in the game, put all of your previous at-bats aside because it's the last one that's very important."
Detroit earned its sixth consecutive win and opened a seven-game lead over Minnesota in the AL Central.
After Rays starter Wade Davis pitched seven sharp innings in his major league debut, Detroit overcame a 3-1 lead when Tampa Bay used five relievers in the ninth. Inge capped the comeback, connecting with one out against Russ Springer (0-3) for his fifth career slam.
"He did a great job," Springer said of Davis. "It makes it worse to ruin his first start. It was bad in a couple different ways. It wasn't that bad of a pitch. It was on the outside corner. I know he's a hook guy, but on paper he's supposed to hit the ball to second. He got it."
Edwin Jackson (12-6) allowed three runs and six hits in eight innings for Detroit. Brandon Lyon pitched the ninth for his third save.
"He gave up that three-spot, but I thought he was tremendous," Tigers manager Jim Leyland said of Jackson. "That's what gave us a chance to win the ballgame."
Evan Longoria hit his career-high 28th homer for the Rays, who have lost eight of 11 and fell seven back of wild card-leading Boston. The 2008 AL Rookie of the Year joined teammate Carlos Pena in reaching 100 RBIs, marking the first time two Tampa Bay players have reached that total in the same year.
Davis allowed one run and three hits with nine strikeouts and one walk. Taken in the third round of the 2004 draft, the native of nearby Lake Wales, Fla., pitched a day before his 24th birthday.
"Everybody wants to win," said Davis, who watched the ninth in the clubhouse. "I gave them a chance."
Davis struck out his first four batters before Aubrey Huff homered to put the Tigers ahead 1-0 in the second. The right-hander then closed out the second with two more strikeouts.
Lance Cormier pitched 1 1-3 scoreless innings, striking out Carlos Guillen to start the ninth. Manager Joe Maddon then brought in Grant Balfour, who walked Miguel Cabrera.
Struggling closer J.P. Howell entered and walked pinch-hitter Marcus Thames, the only batter he faced. Springer took over and gave up a single to Magglio Ordonez before Inge's 27th homer of the season. Randy Choate got the last two outs.
Maddon said the ninth-inning matchups were established before the inning.
"Obviously the two walks can't happen," he said. "The walks really turned that game around."
Tampa Bay's bullpen is 2-8 with eight blown saves in the last 28 games.
Longoria homered in the second and Tampa Bay added two more in the third. Dioner Navarro scored on Carl Crawford's double-play grounder and Fernando Perez came home on Jackson's wild pitch.
Jackson, traded from Tampa Bay to Detroit during the offseason, received his 2008 AL championship ring from the Rays on Saturday. He went 14-11 last season. ... Davis' nine strikeouts set a team record for a rookie in his first game. ... Tigers closer Fernando Rodney, who threw 35 pitches on Friday, rested for the second straight game. ... Leyland plans to talk with RHP Armando Galarraga, whose spot in the rotation might be in jeopardy. ... A group of Detroit players enjoyed watching Leyland drive RHP Jeremy Bonderman's remote-control, high-speed mini-car in the outfield before the game. "We both flipped it," Leyland said.Sign up for breaking news alerts from FOXSports.com!