Garnett, a 12-year NBA veteran, averaged 22.4 points and 12.8 rebounds last season, leading the league for the fourth straight season appearing in 76 games with the Minnesota Timberwolves.
The trade sent Ryan Gomes, Gerald Green, Al Jefferson, Theo Ratliff and Sebastian Telfair to the T-wolves. Minnesota also received a 2009 first-round draft pick and reacquired the conditional first-round draft pick it gave up in the Ricky Davis-Wally Szczerbiak trade.
The Celtics went 24-58 last season.
The deal creates a $54.4 million salary trifecta that includes Garnett, Paul Pierce and Ray Allen, the Boston Herald reported. The Herald said Garnett agreed to a three-year extension on his current contract, which has two years remaining.
The 6-foot-11, 253-pound Garnett is one of just five NBA players to score at least 19,000 career points, 10,000 rebounds and 4,000 assists. He was drafted directed out of Farragut Academy High School in Chicago in 1995 by Minnesota.