Vancouver squeaked out the win after squandering a 2-0 lead in the final 10 minutes of regulation time.
The Canucks (16-10-5) posted their fourth win in five games while the Coyotes (16-8-5) suffered their third loss in four outings.
Higgins scored after he kicked the puck away from Phoenix centre Mike Ribeiro in the Vancouver zone and then raced down left wing and beat Coyotes goaltender Mike Smith with a quick shot to the far side.
Canucks goaltender Roberto Luongo recorded 37 saves for the win as Smith stopped 29 shots in defeat.
The Coyotes rallied in the final 10 minutes of the third period after trailing 2-0 at the end of the second.
Both teams produced 10 shots in a scoreless first period. Luongo stopped David Moss on a rush up the middle in the first five minutes, and Daniel Sedin almost scored while flat on his stomach near the Phoenix net.
Garrison opened the scoring on a power play in the first minute of the second period as he wired in a slapshot from the point less than a minute after Oliver Ekman-Larson took a holding penalty.
Vancouver's power play has now scored in seven consecutive games.
Sedin put the Canucks ahead 2-0 in the final minute of the second period. The play started as Ryan Stanton came out of the penalty box, snared the puck and launched a three-on-two with Daniel and Henrik Sedin.
After botching a pass to Daniel Sedin just inside the Coyotes' blue-line, Stanton made a nice recovery and retrieved the puck off a Phoenix defenceman. Stanton then quickly sent it to Daniel Sedin, who set up his brother for a one-timer that beat Smith cleanly.
Through much of the first 40 minutes, the crowd serenaded Luongo with "Loooo" almost every time he made a save. But the Coyotes never quit working and were finally rewarded for their efforts with a pair of quick goals.
Vermette pulled the Coyotes within a goal midway through the third period as he beat Luongo with a high slapshot from just inside the Vancouver blue-line after the Canucks got lackadaisical with the puck while trying to move it out of their zone. The Phoenix goal came moments after Vancouver's Jannik Hansen missed an open net at the other end of the ice.
Then, just two minutes and 21 seconds later, Moss, standing on top of Luongo's crease forged a 2-2 tie as he put in the rebound off Rob Klinkhammer's shot from the side of net.
Notes: Before the game, a moment of silence was held for late South African president Nelson Mandela, who died Thursday at the age of 95. … Coyotes captain Shane Doan sat out due to illness. Meanwhile, defenceman Derek Morris missed the game after leaving the club to attend to a family matter. … Andrew Alberts replaced defenceman Alex Edler in Vancouver's lineup. Edler is sidelined indefinitely with a lower-body injury.