The victory was just the third in 10 games for the Brewers, who led the National League Central by 4 1/2 games entering Sunday.
The Twins scored four times in the third inning off Claudio Vargas on Michael Cuddyer's two-run homer and a two-run single by Chris Heintz.
The Brewers cut it to 4-2 in their half of the inning on J.J. Hardy's RBI double to score Tony Gwynn Jr. and a big two-out single by Jenkins.
The Twins added a run on a sacrifice fly by Justin Morneau in the fourth for a 5-2 lead, but the Brewers rallied in the fifth.
Hardy led off with a single, and Bill Hall reached on an infield single deep in the hole at shortstop with one out.
Jenkins then hit a 1-0 pitch from Ramon Ortiz that cleared the center-field wall just out of the reach of Torii Hunter, the Twins' centerfielder who has made a career of homer-stealing catches at the wall.
The Brewers took their first lead of the day when Prince Fielder led off the seventh with a double and moved to third on a wild pitch by Minnesota reliever Dennys Reyes.
After Bill Hall grounded out to third, Jenkins delivered by lining an offering from the lefty Reyes into left field over the outstretched glove of third-baseman Luis Rodriguez.
Matt Wise pitched the seventh and earned the victory. Derrick Turnbow pitched a scoreless eighth and Francisco Cordero struck out the side in the ninth to earn his 17th save in 17 opportunities. It was the 150th career save for Cordero.