Last season with Minnesota, Burton was 3-5 with a 4.36 ERA and three saves in 68 appearances. In 64 innings, he permitted 58 hits and 34 runs (31 earned) while walking 25 and striking out 46.
He had his best season in the Majors in 2012, posting a career-best 2.18 ERA in 62 innings. The Twins declined his $3.6 million option in October, instead paying a $200,000 buyout.
With the signing, 68 players are now expected to report to camp. Pitchers and catchers are scheduled to arrive Friday.
Bryan Hoch is a reporter for MLB.com. Follow him on Twitter @bryanhoch and read his MLBlog, Bombers Beat. This story was not subject to the approval of Major League Baseball or its clubs.