The deal was worked out Thursday at baseball's winter meetings in Nashville, Tenn.
The 31-year-old Burton had five saves and a 2.18 ERA in 62 innings with 16 walks and 55 strikeouts in 2012, his first season with the Twins. Burton became the primary eighth-inning setup man for closer Glen Perkins. Burton previously pitched for the Cincinnati Reds.
Burton will make $2.05 million in 2013 and $3.25 million in 2014. The Twins hold a $3.6 million option for 2015 with a $200,000 buyout.