Young scored 17 points in the fourth quarter, when L.A. pulled away and snapped a seven-game losing streak.
Referring to the nickname he gave himself earlier this season, Young told reporters, "I can play a little bit. I just want you all to see that. Swaggy P. can play a little bit. I got a future here. I can do my thing. I've been telling you all. You better watch out for me."
The Lakers trailed by 13 points early in the second quarter, but they used a 19-2 run in the final 3:40 to grab a 57-51 halftime lead.
"We kind of struggled a little bit," said Jodie Meeks, who finished with 23 points. "But that's basketball. We knew we were going to make some runs and Nick got hot. ... We just kind of rode his energy."
With the win, the Lakers improved to 26-55 with one game remaining in their season. The Jazz slipped to 24-57 by losing for the fifth straight time.
"We needed a night like this just the energy," Young said. "We came in here. We knew it was going to be a battle. It was a way to say, 'Who wants to be the last [place] team in the West."
In the final 1:54, Young kept firing. He missed three three-point shots, despite the fact the Lakers owned a 117-100 lead.
"Nick got on a roll," said Jazz coach Tyrone Corbin. "... He's a guy, when he gets on a roll, he's not going to stop shooting it. Even late he was trying to [score]. That's who he is."