Correia, 33, went 5-13 with a 4.94 ERA in 23 starts with the Twins this season. The right-hander struck out 61, walked 32 and allowed 13 homers in 129 1/3 innings, while registering 13 quality starts. It marked the first time in Correia's 12-year career that he was traded.
"It's always tough leaving guys who you played with for a while because they become friends of yours but it's nice to go somewhere and play some important baseball games at the end of the year," Correia said. "It's exciting."
Correia, a San Diego native, will get the chance to play near his hometown for a club leading the National League West. He's expected to give the Dodgers rotation depth and could also be used in long relief.
Correia joined the Twins on a two-year, $10 million deal before last season. He was the club's most consistent starter last year, posting a 4.18 ERA in 31 starts.
"He was nothing but a pleasure," Twins manager Ron Gardenhire said. "Last year, he was our best and most consistent pitcher. This year, I know he had some ups and downs but he always seemed to give us a chance. He was a great guy in the clubhouse and always a pleasure to be around."
Correia said he had no hard feelings about the trade and said he enjoyed his time with the Twins despite their struggles during his tenure in Minnesota.
"I enjoyed every minute of it," Correia said. "The guys on the team and the coaching staff, everyone was great. I wish we would've won more games, obviously. But besides that, I couldn't have enjoyed my time in Minnesota any more."
Milone, meanwhile, was acquired by the Twins from the A's for outfielder Sam Fuld on July 31. The left-hander had a 3.55 ERA in 16 starts with the A's this season and has a career 3.84 ERA in parts of four seasons in the Majors.
The 27-year-old was assigned to Triple-A Rochester after the trade, and made one start there on Tuesday, giving up just one run over seven innings.
"He's a guy who knows how to pitch in the big leagues and has a good track record," Gardenhire said. "We're excited to have him up here. He pitched well in his start at Triple-A."