Boston led by 17 points entering the fourth quarter before LeVert sparked the 129-120 win Tuesday at TD Garden in Boston.
The Nets breakout star made 17 of 26 shots from the field and 5 of 10 shots from 3-point range. He scored 12 of his 51 points at the free throw line.
"I was just thinking about one possession at a time," LeVert told reporters. "The basket [looked] so big at that point."
LeVert scorched the Celtics for 26 points in the fourth quarter. He forced overtime by making three free throws in the final seconds of regulation.
The Nets went on an 8-0 run to start overtime and never looked back. LeVert scored all of the Nets' 11 points in the extra frame.
Brooklyn's win ended a four-game losing streak. Boston has now lost back-to-back games.
"We didn't have any defensive intensity," Celtics coach Brad Stevens said. "I should have subbed a couple of guys out that normally play, and I just kept them in."
DeAndre Jordan scored eight points and had 15 rebounds off the bench for Brooklyn. Spencer Dinwiddie chipped in 14 points in the win. Jaylen Brown led the Celtics with 22 points. Kemba Walker recorded 21 points in the loss. Marcus Smart had 14 points, 10 assists and six rebounds for Boston.
The Celtics (41-19) face the Cleveland Cavaliers at 7 p.m. EST Wednesday at Rocket Mortgage FieldHouse in Cleveland. The Nets (27-33) host the Memphis Grizzlies at 7:30 p.m. EST Wednesday at Barclays Center in Brooklyn.