Kepler had three hits and Josh Donaldson added two for the Twins, who took two of three from the NL Central-leading Cubs.
After winning the AL Central last year, Minnesota clinched its second straight postseason berth on Saturday and trails the Chicago White Sox by two games in the division.
''We had good competitive at-bats all night,'' Twins manager Rocco Baldelli said. ''(Darvish is) good at what he does, but I don't think we made it easy on him tonight. He made mistakes. We hit some balls well and really forced him to work because if you don't force him to work, he'll throw into the eighth inning without much problem or hesitation.''
The Cubs scored just two runs in the series and maintain a 3 1/2-game lead over St. Louis in the NL Central. Chicago had won five of six.
''We're going through a little bit of a stretch where we've won some games but hadn't really lit up offensively,'' Cubs manager David Ross said.
Berrios (5-3) and three relievers combined to limit the Cubs to four hits. The Cubs loaded the bases in the ninth on a passed ball, a hit batter and a walk against Sergio Romo, who struck out Nico Hoerner to end the game.
Berrios struck out four and walked one to improve to 4-0 in his last six starts.
''I turned the page,'' he said. ''I just said, `It's September. New month, new opportunity.'''
Darvish (7-3), an NL Cy Young candidate, allowed a season-high four runs and nine hits in his quest to become the NL's first eight-game winner. He struck out nine and walked one, ending a streak of nine consecutive quality starts. Darvish was 5-2 with a 2.16 ERA in his previous eight starts at Wrigley.
''The fastball was good,'' Darvish said. ''Only the cutter was not good. I couldn't make my cutter better today.''
Kepler chased Darvish with a two-run home run - his eighth of the season - in the seventh.
''It hasn't been the easiest of years,'' Kepler said. ''On and off the field, it's been a lot to adjust to. But tonight I felt like my mind was clear. We've been working in the cage a lot with (hitting coach Edgar Varela).''
Darvish encountered trouble early, allowing three singles and a walk in the first. Eddie Rosario singled to put the Twins ahead before Darvish struck out the side.
Kepler's groundout extended the lead to 2-0 after Jake Cave tripled to lead off the second inning.
The Twins have won three of seven.
Willson Contreras led off the fourth with a double for the Cubs and advanced on a groundout but was thrown out at home on Javier Baez's grounder to third.
Ross declined to explain why left fielder Kyle Schwarber was pulled in the third inning. Schwarber said he misplayed Cave's hit into a triple in the previous inning.
''I don't think the guy should have gotten to third base there,'' Schwarber said. ''I put Yu in a tough spot.''
The Cubs recalled outfielder/infielder Jose Martinez from its alternate site in South Bend, Ind., and optioned left-handed pitcher Josh Osich to South Bend before the game.
Twins: DH Nelson Cruz (sore knee) was out of the lineup for the second consecutive day. With Monday an off day, Cruz - who had been playing with the injury - can take advantage of three days of rest, Baldelli said.
Cubs: RHP Tyler Chatwood (right forearm strain) threw from 105 feet Saturday. Surgery has not been discussed for him, general manager Jed Hoyer said.
ALL IN THE FAMILY
Baez fouled out and grounded into a fielder's choice against Berrios, his brother-in-law, in their first regular-season meeting. Baez flew out twice against him in All-Star Games.
Twins: Minnesota had yet to announce Tuesday's starter to begin a two-game home series against Detroit and LHP Tarik Skubal (1-3, 6.17).
Cubs: LHP Jon Lester (2-2, 4.91) takes the mound Monday to open a four-game series at Pittsburgh. He is 0-2 with a 6.82 ERA in his last seven starts. The Pirates will start RHP JT Brubaker (1-2, 4.79).
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