The Indians will get minor league outfielder Alex Call in the deal.
"We've had a couple of goals going into the offseason, and that's foremost trying to make sure we have a team that's capable of winning the American League Central in 2019 but also position the organization for sustained success beyond that," Indians president of baseball operations Chris Antonetti said. "There are a couple ways to do that. One is to infuse young, controllable talent into the organization. The second element of that is to manage our finances. We feel like the series of moves we've made up to this point advances both of those goals."
Alonso, 31, hit .250 with 23 home runs and 83 RBIs last season for the Indians. A nine-year veteran, Alonso was an All-Star in 2017, when he hit 28 home runs and had 67 RBIs for the Oakland A's and the Mariners.
This was the first trade between the AL Central rivals in more than two decades, and it could have broad implications for the rest of the offseason.
"We haven't aligned on trades for a variety of reasons, but this was one of those instances where what they were seeking to do aligned with what we were seeking to do," Antonetti said.
Chicago already has Jose Abreu at first base, but he played in just 128 games last season because of injuries and could see more time at designated hitter with the addition of Alonso.
"We envision both he and Abreu getting the bulk of the at-bats between the first base and the DH spots through the course of the season. How exactly that breaks down in the end will be a managerial decision," White Sox general manager Rick Hahn said, adding that Alonso and Abreu are friends and Abreu was excited about the deal.
The White Sox also have expressed interest this offseason in free-agent slugger Manny Machado. Alonso is Machado's brother-in-law.
For the Indians, the trade moves Alonso's $8 million salary for 2019 off their payroll, giving Cleveland more flexibility if it wants to keep its deep rotation intact.
Alonso's contract includes a 2020 club option at $9 million with a $1 million buyout. The option becomes guaranteed if Alonso has 550 plate appearances next year or 1,100 in 2018 and 2019 combined and he passes a physical at the end of next season.
"We certainly feel like it improves us in 2019 and like the potential of having him around in 2020 as well," Hahn said.
Call, 24, was a third-round pick of the White Sox in 2016. He was promoted to Double-A Birmingham in June and hit .242 with 12 homers and 58 RBIs in 123 games.
The Associated Press contributed to this report.