Brandon Ingram, who finished the contest with a career-high 49 points, nailed a go-ahead jumper to give the Pelicans a 122-121 lead with 0.2 seconds remaining in regulation.
However, there was more drama to come on the ensuing inbounds play, as Jaxson Hayes was called for a foul after appearing to prevent Jazz center Rudy Gobert from going up for a potential game-winning lob.
Gobert drained one of two free throws to send the game to overtime tied at 122 apiece.
The Pelicans pulled away in the extra frame, posting an impressive victory to end Utah's winning run and notch their fourth win in the last five games.
With his career night, Ingram became the youngest player in Pelicans history to register a 45-point performance at 22 years and 136 days old, according to Stat Muse's Justin Kubatko.
Ingram also became only the third player in NBA history to score at least 49 points on 25 or fewer shots before the age of 23, joining Shaquille O'Neal and Kareem Abdul-Jabbar, according to The Athletic's Will Guillory.
For the Jazz, Donovan Mitchell matched his career-high point total with 46 in defeat.
Gobert fouled out after apparently making contact with Ingram on his drive with 1:19 remaining in overtime. Jazz head coach Quinn Snyder challenged the big man's sixth foul, but the call was upheld after officials went to video review.
The Frenchman took to Twitter postgame to make his feelings known:
Meanwhile, the win improved the Pelicans' record to 16-26. They now sit three-and-a-half games back of the Memphis Grizzlies for the Western Conference's eighth seed.
New Orleans is slated to get back a major reinforcement next week for a potential playoff push, as rookie phenom Zion Williamson is expected to make his NBA debut Jan. 22.