Financial terms were not disclosed, but the Boston Globe reported the deal is worth $30 million.
Damon, 28, becomes the latest addition to a franchise which was sold Thursday for a record $660 million to a group led by Florida Marlins owner John Henry and television producer Tom Werner.
But the sale of the Red Sox has not slowed down general manager Dan Duquette, who signed pitcher John Burkett and acquired second baseman Pokey Reese earlier in the week after previously adding pitchers Darren Oliver and Dustin Hermanson and first baseman Tony Clark.
Boston leadoff hitters hit a league-worst .243 last season, making Damon a perfect fit for a team sorely lacking in speed. He will likely start in center field with Trot Nixon in right and Manny Ramirez in left.
Despite hitting just .256 with 108 runs and 27 steals last season with the Oakland Athletics, Damon told the Boston Globe that Oakland made a strong offer, but he wanted to play closer to his Florida home.
"Oakland did everything in their power to sign me," Damon said. "The biggest thing was moving back to the Eastern time zone. It boiled down to my family."
Two years ago, the left-handed hitting Damon sparkled for the Kansas City Royals, batting .327 and leading the league with 136 runs and 46 stolen bases.
After playing five-plus years with Kansas City and eligible to become a free agent following the 2001 season, Damon was traded by the Royals to Oakland, where he hit only .217 in the first two months of the season and .276 the rest of the way.
The loss of Damon is the latest blow for the Athletics, who were trying to recover from the losses of slugger Jason Giambi and closer Jason Isringhausen. They have added David Justice and reliever Billy Koch.
A career .286 hitter with an on-base percentage of .351, Damon has scored 100 or more runs and hit 30 or more doubles each of the last four seasons. He also has stolen 25 or more bases in five of his six full seasons and has a career .318 average at Fenway Park.
During his career, Damon has consistently ranked as one of the most difficult players to strike out and double up. Last year, he had the third best ratio in the American League with only 70 strikeouts in 719 plate appearances and the fourth best ratio with one double play per 92 at bats.
Notable deaths of 2014 [PHOTOS]