The 33-year-old slugger received 21 of 30 first-place votes and 374 points in voting announced by the Baseball Writers' Association of America.
Cleveland Indians third baseman Jose Ramirez was second with eight first-place votes and 303 points, and New York Yankees second baseman DJ LeMahieu, who won the AL batting crown, followed with one first-place vote and 230 points.
Voting by the BBWAA was completed by the start of the playoffs. It has voted for the award since 1931.
Abreu led the majors with 60 RBIs and 148 total bases, and topped the AL with 76 hits and a .617 slugging percentage. He played in all 60 games during the virus-shortened season as Chicago claimed a wild-card spot.
Surrounded by family members, Abreu put his head down for a minute after hearing he'd won and teared up.
"That was a very special moment,'' he said through an interpreter.
Abreu batted .317 with 19 home runs, connecting six times in a three-game series against the Cubs in late August. That barrage of long balls at Wrigley Field was part of his 22-game hitting streak, the longest in the majors this year.
Abreu gave credit to manager Rick Renteria, who left the team after the season in what was described as a mutual decision. Recently hired Hall of Fame skipper Tony La Russa is now facing charges in a drunken driving arrest; Abreu said he was eager to play for La Russa.
"Keep pushing forward, keep moving forward,'' Abreu said.
Abreu was the 2014 AL Rookie of the Year and is a three-time All-Star. He became the fourth White Sox player to win the AL MVP, joining Frank Thomas (1993-94), Dick Allen (1972) and Nellie Fox (1959).
Abreu was the third Cuban-born player to be an MVP, along with Jose Canseco and Zoilo Versalles.
Smooth around the bag, Abreu ended an MVP drought for AL first basemen. None had won the award since Justin Morneau for Minnesota in 2006; Cincinnati first baseman Joey Votto won the National League MVP in 2010.
Freeman got a $185,185 bonus and Abreu received $37,037 for winning in contract bonuses prorated because of the shortened season.
Ramirez hit .292 with 17 home runs and 46 RBIs. His late-season surge helped Cleveland clinch a wild-card spot. No Indians player has won the AL MVP since Al Rosen in 1953.
LeMahieu led the majors with a .364 batting average. Able to play all around the infield, he is now a free agent.
AL Cy Young Award winner Shane Bieber of Cleveland was fourth and Angels outfielder Mike Trout was fifth. A three-time AL MVP, Trout had finished in the top four every season since he was AL Rookie of the Year in 2012.
The Associated Press contributed to this report.