Patrik Laine and Nikolaj Ehlers scored for the Jets (13-8-1), who have won three straight and are 7-1-1 in their past nine games.
"I thought everyone from top to bottom played real good game," Hellebuyck said. "We stuck together, we grinded hard and we battled. I thought everyone brought their 'A' game tonight."
WATCH: [All Jets vs. Predators highlights]
The Predators (9-8-3) are 1-5-2 in their past eight games.
"We're doing good things," Nashville forward Rocco Grimaldi said. "We had, what, [39-25] shots or whatever it was? We had a lot of pucks to the net, had good opportunities, hit a post. It's just one of those times where it's frustrating, and everyone is frustrated."
Laine gave the Jets a 1-0 lead 18 seconds into the first period when he scored off a pass from Kyle Connor on the rush.
"I think we wanted to have a good start," Laine said. "We wanted to play the first 10 (minutes) hard. Maybe if we don't score, we still want to play hard. So that was a great start. We kind of needed that one. It was a great pass by [Connor] and a pretty easy job to put it in."