Nick Schmaltz scored two goals for the Coyotes (33-29-8) who are 3-5-0 in their past eight games, and three points back of the Minnesota Wild for the second wild card from the West.
Darcy Kuemper made 30 saves for Arizona.
Schmaltz scored twice in the first period, making it 1-0 Coyotes at 15:50 and 2-0 at 19:58. Phil Kessel had an assist on each goal.
"They come hard here at home," Schmaltz said. "They have a lot of skilled players and we didn't close out enough guys. We gave them a little bit too much time and space, and when you do that against high-end players, they're going to make you pay. It's obviously a frustrating loss. Those are big points."
Ehlers cut Arizona's lead to 2-1 at 2:13 of the second period.
At 17:27 of the second period Tucker Poolman tied the game 2-2, scoring on the rebound from an Eakin shot.
"Your 2-0 lead, you've got to really make them earn their goals," Coyotes coach Rick Tocchet said. "And that's basically what it comes down to. We had our chances at the end, but a couple times we were over-passing again. I'm not quite sure, we had a couple of good shots. Keep ripping it. That's what good teams do."
Mark Scheifele scored into an empty net with 36 seconds remaining in the third period for the 4-2 final.