Garnett, who won an NBA title in 2008 with Boston, will lend a veteran presence to a young team in Minnesota, where the 20-year veteran broke into the league as the fifth overall pick of the 1995 draft.
A trio of three-team deals were also made as contenders fine-tuned rosters ahead of the playoffs, while other teams dealt with an eye to the long-term future, according to nba.com as many late deals had not yet been confirmed by the clubs.
Point guard Dragic, who was traded along with his brother Zoran, was averaging 16.2 points, 4.1 assists and 3.6 rebounds in 52 games this season in the last year of his contract.
In another three-way swap, Phoenix acquired guard Brandon Knight from the Milwaukee Bucks, who received guard Michael-Carter Williams from the Philadelphia 76ers and center-forward Miles Plumlee and guard Tyler Ennis from the Suns.
The Sixers, meanwhile, received a protected 2015 first-round pick from the Suns via the Los Angeles Lakers.
Another deadline deal sent Thunder guard Reggie Jackson to Detroit, four players to Oklahoma City including center Enes Kanter from Utah and guard D.J. Augustin from the Pistons, and Kendrick Perkins and a first-round draft pick to Utah.
(Reporting by Larry Fine in New York; Editing by Frank Pingue)