Alex Burrows missed a penalty shot awarded 36 seconds into overtime after he was hauled down from behind by Belanger, shooting wide left.
Hamhuis was on target two minutes later, sliding the puck under Ilya Bryzgalov's pads off assists from both Sedins.
"Hank and Danny [Sedin], anytime they're circling around the puck, as a defenseman you try to get open for them," Hamhuis said. "I was trying to find a spot to get open."
Belanger and Korpikoski combined to tie it at 3 with 43 seconds left in the second period. Belanger flipped a pass from the high slot in front of Korpikoski, who was charging toward the net from the right side. Korpikoski reached the puck just after it touched the ice, chipping it over Roberto Luongo for his first goal since Feb. 25.
"We've got some players who should watch that and see what happens when you go to the net hard," Coyotes coach Dave Tippett said.
Belanger gave Phoenix a 1-0 lead 3:27 into the game. The puck bounced off Salo's skate from the right halfboards to Belanger in the low slot, where he fired a shot off the inside of the left post.
Doan made it 2-0 at 10:55 of the first with help from Belanger. Doan carried the puck behind the Canucks net, passed to Belanger at blue line and skated to the top of the left circle for the return pass and a one-time shot that beat Luongo for a power-play goal.
"I thought our defense could have been tighter," Canucks coach Alain Vigneault said. "We had to come back and we did."
Vancouver cut the lead in half 17 seconds later when Hamhuis scored on a long rebound from the middle of the right circle.
Henrik Sedin tied it on the power play at 9:14 of the second when the rebound of Ryan Kesler's shot from the right circle dropped into the crease and onto Sedin's stick.
The Canucks took their first lead at 12:17 of the period as Salo fired a shot from the right point off Coyotes defenseman Michal Rozsival in front of the net.
"We've had some bad breaks go against us," Tippett said.
Coyotes D Rostislav Klesla left the ice at 3:20 of the second after the puck bounced off the heel of his stick and hit him in the face. Klesla skated off under his own power with a towel to his forehead. He returned for one shift in the third period before returning to the bench for good. ... The Sedin brothers have 17 goals and 20 assists in a combined 75 career games against the Coyotes.