Athletics officials didn't release the value of the contract but multiple reports estimated it to be worth $10 million plus incentives.
Sheets, who spent the first eight years of his career with the Milwaukee Brewers, missed all of last season while recovering from the operation. Scouts watched as he threw in Louisiana recently and reportedly were impressed.
Published reports indicated the New York Mets and Texas Rangers were also competing for Sheets's services.
"The elbow is doing great. The biggest thing that I'm going to take out of this past year, that I didn't play, is that my whole body feels a lot better," Sheets told reporters at an introductory news conference. "Baseball is such a grind, such a long season, that you kind of forget how good you can feel. Missing this whole year, and concentrating on everything, I feel renewed."