The Timberwolves also received cash considerations in the deal that was completed Wednesday night.
The trade will save the Knicks $1.8 million and give the Wolves a center to audition for the rest of the season. Milicic has been mired at the end of Mike D'Antoni's bench for months and has publicly said he's likely to return to Europe to play basketball when his contract expires this summer.
"He understandably was frustrated and he may feel that way now," David Kahn, Minnesota's president of basketball operations, said at halftime of the Timberwolves' game at Washington.
"I'm hopeful he clears his mind of that for the next couple of months," Kahn said.
The Knicks plan to waive Cardinal in an attempt to clear another roster spot to facilitate a potential deal that would send Nate Robinson to Boston.
The 24-year-old Milicic was selected by the Detroit Pistons with the second overall pick in the 2003 draft, just behind LeBron James and ahead of Carmelo Anthony, Chris Bosh and Dwyane Wade. He is heading to his fifth NBA team.
Milicic has played in just eight games this season for the Knicks. He is averaging just 2.0 points and has not played since Nov. 13. Kahn hopes he doesn't bolt to Europe.
"I hope he will give it one more shot," Kahn said. "Even if he ends up returning to Europe, I hope he wants to end his NBA career on a positive note. In our situation, he'll play."
The Timberwolves entered Wednesday's game with a 13-41 record.
"There's a lot of enthusiasm among our coaches," Kahn said. "Let's see what he can bring."
The 32-year-old Cardinal is averaging 1.7 points in 27 games this season. He has received sparse playing time for the young and rebuilding Timberwolves.
Chad Ford covers the NBA for ESPN Insider. Information from The Associated Press was used in this report.