Anthony, who was ejected during Friday's loss to the Chicago Bulls, finished with 33 points, scoring 19 in the fourth quarter, as the Knicks came from behind to secure the home win.
"We didn't want to look back at this game later in the season and say this was a game we shouldn't have lost," Anthony told reporters. "We picked it up defensively in the second half and we won the game."
New York trailed for virtually the entire game before Anthony went on an 8-0 run in the final two minutes to put the Knicks up by four.
Anthony scored the Knicks' last 12 points to seal the win.
"That was kind of an MVP performance at the end. He stepped up and made the plays," said New York coach Mike Woodson. "When he got that fifth foul called, it's like a light went off and he made the plays big-time."
J.R. Smith scored 19 off the bench for New York (20-7) while Tyson Chandler had 16 and nine rebounds as the Knicks maintained their 5 1/2 game lead in the Atlantic Division.
Nikola Pekovic recorded 21 points and 17 rebounds to lead the Timberwolves (13-12), who were missing All Star Kevin Love due to an eye ailment.
Minnesota went up by nine at half-time and led by 11 midway through the third before New York made their move.
Chandler scored 10 in the third quarter as the Knicks cut the deficit to two heading into the fourth.
New York are still without forward Amar'e Stoudemire, who has been out the entire season with a knee injury but has recently started practicing with the team.
Despite his absence, the Knicks are off to a strong start and have a 5 1/2-game lead in the Atlantic Division.