Zucker made it 1-0 with a power-play goal at 17:41 of the first period with a one-timer from the right point.
Arizona has four power-play goals in its past two games, including three in a 6-2 win against the St. Louis Blues on Thursday.
"I thought it was a good period of hockey and then we take two penalties late in the first and they generated some energy off that," Anaheim coach Greg Cronin said. "I thought that changed the complexion of the period and that changed the complexion of the game."
Keller made it 2-0 at 14:01 of the second with a wrist shot from the right circle on a delayed penalty call, with Arizona having an extra skater. The Coyotes controlled the puck in the offensive zone for 1:01 before Keller's shot went past a screened Dostal.
"They had a tired crew on the ice and they're pushing to change, and we had a really good possession," Tourigny said. "That opened up play for us, [we] had traffic in front of the net and 'Kells' put one right in the hole."
Vatrano cut it to 2-1 with a power-play goal at 7:47 of the third, driving to the net and scoring off Jakob Silfverberg's cross-ice pass.
"Before the game, we kind of highlighted some things we wanted to do on the power play, generate a lot of chances and we were lucky to get out there with some time left and capitalize," Vatrano said "[But] we could have had a little more traffic in front of [Vejmelka's] eyes, made it a little harder on him."