After Alex Avila grounded out and Jhonny Peralta was retired on a fly ball, Inge lined a 2-1 offering from Glen Perkins (4-4) over the left field wall to give Detroit two straight wins to open this three-game series.
"I like situations like that, I really do," said Inge. "You're not always going to come through, but there's something about those situations, knowing if you get the result how spectacular it is."
Wilson Betemit also had a solo homer for the AL Central-leading Tigers, who have won eight in a row overall and remained 9 1/2 games ahead of Chicago for first place after the White Sox beat Cleveland in extra innings Saturday.
Joe Mauer homered and had two hits for the Twins, who have lost seven of their last eight and fell to 4-16 in their last 20.
"Our bullpen did a good job. Perk made one bad pitch," said Twins manager Ron Gardenhire. "He got a fastball up, middle in, and Brandon was trying to do one thing. He got a fastball and didn't miss it."
Both starters fared well, but left with no-decisions.
Minnesota's Anthony Swarzak remained winless since August 3 after allowing two runs on four hits and two walks in six innings. He struck out six.
Betemit homered to right on a 3-2 changeup with one out in the third inning for a 1-0 Tigers lead, but Mauer led off the fourth with a blast to center to tie the game.
Andy Dirks hit a ground-rule double to center leading off the bottom of the fourth and scored on Victor Martinez's two-out single to make it 2-1. The Twins tied the game again, 2-2, in the sixth when Trevor Plouffe drew a leadoff walk, went to third on Mauer's single and scored on Danny Valencia's sacrifice fly.
The Tigers lead the season series 13-4...Swarzak is 0-3 in his last seven starts...Detroit hasn't won a division title since capturing the East in 1987.