Abreu went deep to left center on the first pitch he saw from Jorge Alcala (0-1) for his seventh homer. He narrowly missed another one in the third when Max Kepler reached over the right-center wall to catch Abreu's drive. The AL MVP had two hits and scored three runs.
Yasmani Grandal lofted a three-run homer in the second for first-place Chicago.
It was just the sixth multi-homer game for the White Sox, who are missing slugging young outfielders Eloy Jimenez and Luis Robert with long-term injuries.
No matter. The White Sox are rolling.
''I'm getting spoiled,'' manager Tony La Russa said. ''We're going every night. Our offense up and down the lineup did the job. Then the big outs by the bullpen.''
Garrett Crochet (1-2), the second of five Chicago relievers, worked two-thirds of an inning for his first career win. The 6-foot-6 lefty was reinstated from the IL last Friday after recovering from an upper back strain.
''I've been looking forward to this moment for a while,'' said Crochet, who worked on his mechanics while injured. ''I was ready to go on the attack from the moment I went on the mound.''
Tim Anderson and Adam Eaton each had two hits and drove in insurance runs in the seventh. Leury Garcia and Nick Madrigal added RBI singles in the eighth for Chicago.
''That's what you've got to like about our offense, with the big guys missing,'' La Russa said. ''The way we competed through the eighth inning is a good sign.''
Jorge Polanco had a solo shot among two hits for Minnesota, which has dropped two straight and five of six. Nelson Cruz, Willans Astudillo and Andrelton Simmons each had two hits.
''I think our guys have to pick each other up and there's only so much that I can say or the staff can say to get guys in the frame of mind and it is frustrating,'' Twins manager Rocco Baldelli said. ''It works when your teammates are the ones that are motivating you and keeping everything in line and going.''
White Sox starter Dylan Cease, coming off two dominant outings, labored at times in allowing three runs on five hits in five innings. He struck out seven, walked two and hit two batters.
Minnesota's Kenta Maeda allowed three runs on four hits in five innings.
The Twins took a 3-0 lead in the second, ending Cease's streak of 14 scoreless innings.
Polanco went deep to right on a 1-2 pitch down the middle with one out.
Trevor Larnach walked after his catchable foul pop dropped between Grandal and Yoan Moncada off the third-base line.
After Astudillo and Simmons each singled to load the bases, Cease hit Ben Rortvedt to force in a run and Luis Arraez followed with a sacrifice fly.
Grandal's homer to right tied it at 3. Then Cease recovered to get nine more outs and keep the game knotted.
''That was one of those games where it was kind of a grind,'' Cease said. ''I let it get away from me a little, but finished strong and the offense picked us up.''
HITCH IN THE ROTATION
White Sox starters had allowed only one earned run and no homers over 29 innings in Chicago's previous five games.
Minnesota reinstated Luis Arraez from the seven-day injured list (concussion-like symptoms). He led off and started in left field.
To make room on the roster, the Twins optioned INF Nick Gordon to Triple-A St. Paul.
Twins: Baldelli said rookie 1B/OF Alex Kirilloff is undergoing treatment for his right wrist sprain and may start swinging a bat within a few days. Kirilloff got hurt sliding into second on May 3.
Twins LHP J.A. Happ (2-0, 1.99 ERA) faces White Sox LHP Dallas Keuchel (1-1, 3.79) on Wednesday night.
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