The contract calls for Crain to make $4 million in 2011, and $4.5 million each in 2012 and 2013. Crain said the Twins didn't make an effort to keep him before he signed with Chicago.
"It's going to be weird," he said Monday. "I came up with the Twins. I've never been anywhere else. I'm very good friends with the players over there. [So] I'm going to look at it as a challenge. It's going to be fun to go back there and face them."
Crain will join the back end of a White Sox bullpen that was left with a pair of major vacancies when closer Bobby Jenks was non-tendered and main setup man J.J. Putz left via free agency to join the Arizona Diamondbacks.
Crain, who has never been used as a closer, went 1-1 with the Twins last season, while posting a 3.04 ERA. The right-hander had a 0.28 ERA in a 35-game stretch from June 10-Aug. 3, including 21 consecutive scoreless appearances at one point.
Crain said Monday, though, that he has been told that he will get a chance to finish off victories in what is shaping up as a closer-by-committee situation.
"From what [general manager] Kenny Williams said, they have [Chris] Sale and [Matt] Thornton -- two lefties who throw hard at the end of the game," Crain said. "And they have me and [Sergio] Santos. I will be the right-handed guy at the end of the game and mix in with Sale and Thornton. Depending on how it goes, I will have a chance to save some games."
The Toronto native and Houston resident has a 3.42 career ERA in 376 relief outings, playing all seven of his major-league seasons with the Twins, who drafted him in the second round in 2002. He has just three career saves.
Crain, who will wear uniform No. 26 with the White Sox, is 20-7 with a 2.82 ERA in his career against AL Central opponents.
Crain had been talking to Colorado and Boston before the White Sox made an offer last week.
"It actually came together really quick," he said. "We were contacted by a lot of teams after I became a free agent ... We had negotiating a lot with the Rockies and some with the Red Sox and [the White Sox] called on a Tuesday [and] we agreed to terms on Wednesday."
In the meantime, Crain said he's not sure how the Twins will fill a hole caused by his departure.
"I'm not exactly sure what direction they're going. They've always been a team that's built from within," he said. "They like to find guys and bring them up through their farm system. With that being said, I'm not sure who's going to come up."
Doug Padilla covers the White Sox for ESPNChicago.com. ESPNChicago.com baseball reporter Bruce Levine and The Associated Press contributed information to this report.