Detroit was playing its first game since Miguel Cabrera suffered a season-ending biceps injury in Tuesday's series opener. His replacement, John Hicks, had three hits.
With the Tigers down 2-1, Hicks got the rally started with a one-out single. Niko Goodrum and Grayson Greiner followed with singles off Addison Reed (1-5) to tie the game.
One out later, Reyes ripped his single to left to knock in Goodrum. Leonys Martin added an RBI single and another run scored on a throwing error.
Goodrum had three hits, including a solo homer, and scored twice. Reyes and Martin added two hits apiece.
Joe Jimenez (3-0) tossed an inning of scoreless relief and was credited with the victory. Shane Greene got the last three outs, including a pair of strikeouts, for his 16th save.
Eduardo Escobar had two hits and drove in both Twins runs.
The starting pitchers each wound up with no-decisions. Minnesota ace Jose Berrios gave up one run on seven hits and struck out seven in six innings. Detroit's Matthew Boyd allowed two runs on three hits and struck out five in five innings.
The Twins struck quickly. Brian Dozier was hit by a pitch leading off the game and advanced to third on Eddie Rosario's double. Escobar brought them both home with a single to center before Boyd recorded his first out.
Goodrum got the Tigers on the board in the second when he crushed his sixth homer over the right field wall.
Berrios escaped a bases-loaded jam in the fifth when Victor Martinez grounded out into the shift.
Detroit had runners in scoring position with no outs the next inning but still couldn't push across the tying run. Berrios struck out the next two batters and retired Reyes on a flyout.
Escobar tripled to right-center with one out in the eighth but was left stranded as Jimenez set down the next two batters.
--Field Level Media