Palat gave New Jersey a 1-0 lead at 3:58 of the first period when he took a shot from the edge of the right circle that deflected off the stick of Vegas defenseman Kaedan Korczak and over Thompson's glove.
It was Palat's first goal since returning from a groin injury on Jan. 5 after missing 32 games.
"It feels pretty good," Palat said. "I had a couple of chances in those previous games, but it just feels good to contribute and score."
Hutton tied it 1-1 with a wrist shot from the left circle at 12:00 of the second period, and Karlsson made it 2-1 at 14:51 when he poked in a loose puck behind Vanecek.
The Devils outshot the Golden Knights 4-0 in overtime.
"I haven't been good in my last 10 games," Thompson said. "I've got to start winning hockey [games]. That's what I'm here to do, and I haven't done my job. I can make the save on all those goals."