Detroit won its sixth in a row 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 tried five relievers in the ninth. Inge capped the comeback, connecting with one out against Russ Springer (0-3) for his fifth career slam.
Edwin Jackson (12-6) allowed three runs and six hits in eight innings. Brandon Lyon pitched the ninth for his third save.
Evan Longoria hit his career-high 28th homer for the Rays, who have lost eight of their last 11. 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 mark in the same year.
Davis allowed one run, three hits, one walk and struck out nine. Taken in the third round of the 2004 draft, the native of nearby Lake Wales, Fla., pitched a day before his 24th birthday. He had 51-38 record in the minors.
Davis struck out his first four batters before Aubrey Huff homered to put the Tigers ahead 1-0 in the second. The Rays 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 with a 3-1 lead. Tampa Bay manager Joe Maddon replaced Cormier with Grant Balfour, who walked Miguel Cabrera with one out in the ninth.
Struggling closer J.P. Howell then entered and walked pinch-hitter Marcus Thames, the only batter he faced. Springer took over and gave up a single to Magglio Ordonez to load the bases before Inge followed with his 27th homer of the season.
Randy Choate got the last two outs.
Longoria homered in the second. Tampa Bay took a 3-1 lead in the third when one run scored on Carl Crawford's bases-loaded double-play grounder and a second on a wild pitch by Jackson.
Notes
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, was rested for the second straight game. … Detroit manager Jim 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.