Anthony had 30 points, nine assists and seven rebounds, leading the Knicks to a 118-111 overtime victory over the Jazz on Wednesday night.
Robin Lopez added 22 points and 12 rebounds for New York, which trailed by as many as 12 in the second half and by eight entering the fourth quarter. Derrick Williams scored 18 points, and rookie Kristaps Porzingis had 16.
"We stayed composed," Anthony said. "We chipped away and we buckled down on defense. We committed ourselves on defense, and we found energy and momentum."
The Jazz trailed 99-96 after Porzingis made two free throws with 9.9 seconds left in regulation. But Porzingis committed his sixth foul on Hayward's 3-point attempt with 2.4 seconds to go, and the Utah forward hit all three free throws.
"There were a couple of unnecessary fouls for me that I can avoid and I've got to learn from those," Porzingis said. "I've got to learn from my mistakes. It's frustrating to sit on the bench and in overtime."
The game went into overtime after Anthony missed a runner at the buzzer.
Rudy Gobert gave the Jazz its final lead when he opened the extra session with a dunk. New York responded with a 9-2 spurt that included a tiebreaking 3-pointer for Williams.
"I thought a level of grit and mental toughness is required to persevere in those situations," Knicks coach Derek Fisher said. "I thought our guys, kind of regardless of what happened, kept playing and found ways to make the necessary plays to win."
Hayward's 3-pointer with 32 seconds left trimmed New York's lead to 115-111. Following a turnover by Langston Galloway, Hood misfired on a 3-pointer and Utah did not score again.
"I thought tonight we controlled and outplayed them the entire game and then in the last six minutes we kind of fell apart," Hayward said. "We're a young team. We need to learn from these and take positives from it, but it's definitely frustrating."
"We played a great first half and we talked about what the game would become in the second half," coach Quinn Snyder said. "They were going to raise their physicality and they did. The game got very, very physical. We have to be tougher."
Jazz: The Jazz had beaten the Knicks three straight times, including a 2-0 sweep last season.
Knicks: The Knicks' six-game home winning streak is their longest since they won their final 10 home games of the 2012-13 season. ... Lance Thomas (sore left knee) missed his first game of the season.
Anthony currently holds a slim lead in All-Star voting for the third frontcourt starting spot in the East, 368,336 to 361,307, over the Pistons' Andre Drummond. The starters will be announced on Thursday.
Utah is in the midst of a four-game road trip against Eastern Conference opponents. The Jazz is 8-7 against the East this season.
Jazz: Visit Brooklyn on Friday.
Knicks: Host the Los Angeles Clippers on Friday.