Teixeira will rest the wrist completely for four weeks before beginning rehab work. His projected return would be sometime in May.
The injury, originally thought to be a strained forearm, occurred Tuesday while Teixeira was hitting off a tee in Glendale, Ariz., before an exhibition game. He immediately was removed from the Team USA roster for the upcoming World Baseball Classic.
With Teixeira sidelined, the Yankees' options at first base include Juan Rivera, Dan Johnson and Kevin Youkilis.
Teixeira's injury means the Yankees will also begin the season with third baseman Alex Rodriguez and outfielder Curtis Granderson -- who combined with Teixeira will make $62 million this year -- on the disabled list.
Two other Yankees stars -- shortstop Derek Jeter and reliever Mariano Rivera -- are coming back from injuries in 2012.