The deal was announced Monday. The Twins assigned Nunez to Triple-A Rochester.
Signed by the Yankees in 2004, Nunez came up to the major leagues in 2010 and was seen as a possible successor to Derek Jeter at shortstop. But he struggled defensively, was beaten out by Yangervis Solarte during spring training for a backup infield job and was designated for assignment last week.
The 26-year-old Nunez hit .267 in 270 games for the Yankees with 10 homers and 76 RBIs.
The 20-year-old Sulbaran was 9-4 with a 2.96 ERA last year at Class A Great Lakes and Cedar Rapids. He is 21-10 with a 3.15 ERA in 45 starts and 10 relief appearances in the minor league systems of the Twins and Los Angeles Dodgers.