There was more fire and pep in the Isles in the second period and Beauvillier evened the score 34 seconds into the frame, potting his own rebound after being denied by Antti Raanta off a rush.
The Coyotes were outshooting the Islanders 23-9 by the time Brendan Perlini deflected a shot past Jaroslav Halak at 9:52. After that, the Islanders came alive, outshooting the Coyotes 25-10 on the back nine.
Beauvillier tied it 2-2 early in the third, converting a Thomas Hickey rebound past Raanta at 2:53. Beauvillier's goal was his eighth in his last seven games, marking the most productive stretch of his NHL career.
The Islanders looked to be in control, but couldn't beat Raanta for the go-ahead goal. Their best chance was in overtime, when Nick Leddy got a high wrister off within 15 feet and he and captain John Tavares took a pair of whacks at a loose rebound. When they couldn't convert, the Coyotes went back the other way, with Cousins pulling the winner around Halak.