The Twins built a 6-1 lead, but then gave up two big innings in a game that featured 26 hits.
Inge opened a four-run fourth with a homer and Magglio Ordonez finished it off with a two-run single that brought the Tigers within 6-5.
Detroit loaded the bases with one out in the sixth thanks to a fielding error by center fielder Denard Span and Miguel Cabrera was hit by a pitch to drive in the first run of the inning.
Boesch then doubled home two runs and Inge followed with another two-run double. Inge scored the sixth run in the inning on another double by Ryan Raburn.
The loss went to Ron Mahay (0-1), the second of three pitchers who worked during the sixth.
Jim Thome and Luke Hughes hit homers to help the Twins build their early five-run lead.