Scheifele tied his NHL career-high in goals with his 32nd at 15:48 when he scored on the power play to pull the Jets to within 2-1. Connor tied it 2-2 when he got to a loose puck and scored from the slot at 18:21.
Wheeler scored with a wrist shot off a rush down the right side to give the Jets a 3-2 lead at 14:09 of the third period, moments after Mikael Granlund, making his Predators debut, whiffed on a shot attempt on a 2-on-1 rush.
"I think, at times, we kind of shot ourselves in the foot with being a little bit aggressive, trying to push for that next one, push for that next one," Predators defenseman Ryan Ellis said. "If we get that one, we're obviously in a comfortable spot. Tonight, we didn't. They counter and they buried."
Tyler Myers scored into the empty net at 18:14 to make it 4-2, but Ekholm brought the Predators back to 4-3 at 19:00.
Hayes scored his first goal with the Jets into the empty net at 19:28 for the 5-3 final. He was acquired in a trade from the New York Rangers on Monday.