The deal was announced Tuesday during baseball's winter meetings.
Slowey went 0-8 with a 6.67 ERA this year and was out from May to August because of an abdominal strain. He made eight starts and six relief appearances and the Twins went 0-14 in those games.
Slowey became the first major league pitcher to make at least eight starts and lose them all since Lou Sleater of the St. Louis Browns in 1951, STATS LLC said.
The 27-year-old Slowey is 39-29 with a 4.66 ERA in five seasons with the Twins. He's been one of the best control pitchers in the majors over that span.
Slowey has averaged just 1.42 walks per nine innings, the second-lowest rate in the majors since 2007. Roy Halladay is the only pitcher with a better mark at 1.41.
Slowey went 13-6 in 2010 when the Twins won their second straight AL Central title. Minnesota plummeted to last place this season and lost a league-high 99 games.