The 36-year-old Damon brings another left-handed bat to the Tigers' lineup besides switch-hitter Carlos Guillen.
Damon also was courted by the Chicago White Sox and Atlanta Braves since leaving the World Series champion New York Yankees as a free agent.
Tigers general manager Dave Dombrowski has said Damon wasn't in the team's plans as late as January. But Damon came into the picture after the Tigers dealt leadoff hitter Curtis Granderson to the Yankees, then allowed Placido Polanco to become a free agent.
"We needed to make some adjustments," Dombrowski said. "We aren't doing it only for this year, but we also needed to move forward. The moves we made earlier in the year allowed us to make some adjustments, not only for now but also for the future. We also brought in some young talent."
Damon hit .282 with 24 homers for the Yankees last season and is expected to lead off for the Tigers.
"I'm happy for him that he's found out where he's going to go," Yankees manager Joe Girardi said Sunday. "I think for players that's an anxious time. Trying to figure out where you're going to be. Usually it doesn't go on this long. I wish Johnny luck. Johnny was good for us here. We all miss Johnny. This is the hard part of the game when those relationships change a little bit. I'll root for Johnny except when he's playing against us."