The 38-year-old Happ will join a Twins rotation that includes Kenta Maeda, Jose Berrios and Michael Pineda. Minnesota finished third in the AL last season with a team ERA of 3.58.
Happ will become only the second off-season addition by the Twins, who previously added reliever Hansel Robles. Pitchers Jake Odorizzi and Rich Hill became free agents.
The one-time All-Star selection posted a 2-2 record and 3.47 ERA in nine starts last season with the Yankees. He went 12-8 across 30 starts in 2019.