Cabrera's tiebreaking homer off hard-throwing Neftali Feliz (1-1) was a 399-foot shot that landed in the Rangers bullpen in right-center. Inge followed with a blast to left, his second homer of the game.
Inge's two-run shot in the fifth, his first of the season, put Detroit up 6-1.
Phil Coke (3-0), the third Tigers pitcher, worked 1 2-3 shutout innings before Jose Valverde pitched a perfect ninth against the top of the Texas lineup for his sixth save in seven chances.
Coke took over with one out in the seventh after the Rangers had loaded the bases off Joel Zumaya. The left-hander struck out Josh Hamilton on three pitches before giving up a two-run single to Vladimir Guerrero that that tied it at 6.
David Murphy had a two-run double and a run-saving catch after he entered the game for Texas in the top of the sixth for right fielder Nelson Cruz, who reaggravated a right hamstring that has bothered him for the past week. The Rangers said Cruz is day to day.
Detroit, which wrapped up a 5-6 roadtrip, led 4-0 after the first four batters of the game reached and scored off Matt Harrison.
Austin Jackson, the rookie center fielder playing not far from his hometown in Denton, had a leadoff single before consecutive walks. Cabrera had an RBI single, then Ryan Raburn cleared the bases with a one-out double -- a ball high off the 14-foot wall in left-center field that would have been a grand slam if only a few feet higher or to the right where the height of the wall decreases to eight feet.
Tigers starter Jeremy Bonderman retired the first 11 batters he faced until Hamilton homered in the fourth to make it 4-1. After getting his first day off Sunday, Hamilton's second homer of the season ended a 6 for 36 slide and he added a single and scored again in the sixth before his strikeout in the seventh and a game-ending lineout.
Hamilton and Guerrero had consecutive two-out singles in the sixth before David Murphy's two-run double made it 6-4.
Murphy's defensive gem ended the eighth when he sprinted into the right-center field gap and extended to grab Johnny Damon's liner. Damon had already extended his hitting streak to 13 games with a single in the second.
A wild pitch by Bonderman in the fifth, very high and outside, allowed a run after Cruz had a leadoff double and moved to third on Justin Smoak's double that was his first major league hit.
Bonderman struck out three, walked one and gave up six hits in 5 2-3 innings.
Harrison allowed six runs over six innings, with five strikeouts and three walks.
NOTES: Smoak drew a walk in the sixth, the last batter faced by Bonderman. That gave Smoak at least one walk in each of his first four career games, a Rangers record and the first player in the majors to do that since Evan Longoria for Tampa Bay in 2008. ... Tigers SS Adam Everett (mild right hamstring strain) missed his second straight game. ... Tigers manager Jim Leyland said RHP Zach Miner, who has been on the disabled list with a right elbow problem since spring training, is throwing off a mound and "is about ready to pitch."