Ducks defenseman Cam Fowler hit the crossbar seconds before Scheifele scored.
The Ducks led 3-1 after two periods.
Wheeler scored at 11:35 of the third period to make it 3-2 after Nikolaj Ehlers made a centering pass in front. Byfuglien tied the game after charging to the net and muscling the puck over the goal line.
"We decided to not just quit," Ehlers said. "We wanted to keep going and get the two points no matter that. We worked our [butts] off those last 20 minutes. It feels good now."
Armia scored his fourth shorthanded goal of the season for the Jets, who were eliminated from Stanley Cup Playoff contention on Monday.
Silfverberg scored his 22nd at 8:17 of the first period to make it 1-0. Ryan Kesler won a draw back to Silfverberg, who spun off his check and got a wrist shot off from the middle of the left circle, beating Hutchinson over his shoulder.
Armia tied it 1-1 with his 10th goal at 11:54 on a quick shot down low.
Perry made it 2-1 at 18:56, backhanding a rebound past Hutchinson.
Perry scored his second of the game on the power play at 7:01 of the second period to make it 3-1, moving the puck in off the half boards and putting his wrist shot under Hutchinson.