The Ducks scored power-play goals on two of their first three shots to take an early 2-0 lead.
Terry intercepted a clearing pass up the wall, skated into the right circle and scored with a wrist shot for a 1-0 lead at 3:46 of the first period.
He made it 2-0 at 9:35 on a rebound, one second after a 4-on-4 ended.
"I think the worst thing is we get up 2-0 and we're thinking, 'OK, this might be easy,'" Anaheim coach Greg Cronin said.
Crouse cut it to 2-1 at 11:41 on a one-timer after a feed from J.J. Moser (two assists), and Cooley tied it 2-2 at 14:51 when he took a pass from Moser as he entered the Ducks zone with speed and scored on a breakaway.
"With the way they play, if you're not sharp and you're not skating, they're going to get quality chances and they scored on two of them in the first period," Cronin said. "We're lucky there wasn't more."
Crouse scored again at 14:25 of the second period when he won a battle for position in front of the crease and made it 3-2 on a backhand after Matias Maccelli sent the puck toward the net from above the left circle.
"We had a tough start. We battled back. We came back in the game, and we even had a lead," Coyotes coach Andre Tourigny said. "We could expect to win in that type of situation."