Both teams announced the move Thursday. Quintana leaves the South Side after six seasons with the White Sox. The 2016 All-Star posted a 3.20 ERA and a 13-12 record in 32 starts last season. He is 4-8 with a career-high 4.49 ERA through 18 starts this season.
The White Sox acquired outfielder Eloy Jimenez, right-handed pitcher Dylan Cease, first baseman Matt Rose and infielder Bryant Flete in the deal.
Quintana, 28, is under contract through 2020 with $1 million buyouts starting in 2019. The left-handed pitcher is making $6 million this season and is due $8.35 million in 2018.
"It is always extremely difficult to trade a person and player like José Quintana," White Sox general manager Rick Hahn said in a team statement. "This deal moves us closer to our goal of building a team capable of contending for multiple championships over an extended period of time."
The White Sox (38-49) haven't made the playoffs since 2008.
Sources told Fox Sports earlier this week that the Minnesota Twins were "checking on controllable starters." That list included Quintana.
Jimenez is baseball's No. 5 prospect, according to Baseball America. Cease is ranked No. 83.
The White Sox now have the No. 1, No. 8, No. 11, No. 23, No. 28, No. 36, No. 59, No. 63 and No. 68 ranked players of the Top 100 prospects in baseball, according to MLB Pipeline.
Cease is No. 63 on that list. Those nine prospects in the Top 100 are tied with the Atlanta Braves for the most in baseball.
Cubs starters have the eighth-highest ERA in the National League (4.66).