Alex Goligoski tied the game 3-3 for Arizona on a shot from the point that deflected off defenseman Hampus Lindholm at 10:38 of the third period.
The goal came 38 seconds after Troy Terry scored on a wrist shot from the right face-off circle on the rush to give Anaheim its first lead, 3-2, at 10:00. It was his first goal since Oct. 29.
"The last six or eight games, I've felt really good. It hasn't been a lack of confidence, it's mostly just been just a frustration thing," Terry said. "It's hard when you're not getting on the score sheet and you feel like you're playing good hockey."
Coyotes coach Rick Tocchet said he liked the response to Terry's goal.
"They scored that goal, and everyone got dejected, but [Goligoski] goes out and he just pounds it, and it goes off whoever is in front," Tocchet said. "That's the stuff that you need, just pound the puck at the net. Sometimes we defer too much."
Dvorak redirected Oliver Ekman-Larsson's shot from the point on the power play to give the Coyotes a 1-0 lead 3:02 into the game.
"I've been working on tips, things like that -- that's where goals are scored, so that's where I try to get to," Dvorak said.
Brendan Guhle scored with a shot from the point through a Terry screen that went in off defenseman Ilya Lyubushkin's knee to tie it 1-1 at 6:22.
Dvorak put Arizona ahead 2-1 at 18:22 when he scored off a Vinnie Hinostroza pass that was blocked before it got through to Derek Stepan to the left of the crease. It was his second multigoal game of the season (two goals against Winnipeg Jets on Oct. 15).
Sam Steel scored on the rebound of Fowler's shot from the left point to tie it 2-2 at 11:42 of the second.
Gibson extended his right pad to make the save on Michael Grabner's shot toward an open net at 16:07 of the second.
"It was a fun game -- they play a fun style and they've got some skill," Terry said. "They made some plays, but our goalie was pretty unbelievable a couple times."
The Coyotes had two shots on goal during a power play in overtime after Max Jones was called for slashing Stepan at 2:41.
Dvorak and Conor Garland scored in the four-round shootout for Arizona, and Ondrej Kase scored for Anaheim.
Coyotes forward Phil Kessel played in his 800th consecutive game, the third-longest streak among active players and the seventh longest in NHL history.
