Bieksa chipped his fourth of the season over Phoenix goalie Mike Smith just as a Coyotes power play expired in the extra period.
Roberto Luongo, who was shaky at times, made 29 saves for the Canucks.
Smith made 20 stops for the Coyotes.
It was a night to forget for both Luongo and Smith as two of the three goalies heading to next month's Winter Olympics for Canada fought the puck and looked lost in the crease at times.
After a wild finish to the second period that saw Phoenix trail 4-3, Doan tied the game on a breakaway at 5:24 of the third. The Coyotes captain moved in on Luongo and lost control of his deke but still saw the puck squeak through the five-hole for his 15th of the season.
The Canucks and Coyotes both had a couple more chances as the period wore on, with both goalies having to be sharp on deflections in front, but neither could find a winner through 60 minutes of action.
Vancouver was handed a power play right at the end of regulation when Phoenix defenceman Derek Morris was whistled for high sticking, but the league's 28th-ranked unit failed to convert in overtime before Bieksa won it.
Down 2-1 in the second, the Coyotes tied it while shorthanded against Vancouver's feeble 28th-ranked power play. Canucks forward Ryan Kesler tried to slap the puck off Vermette's stick in front of Luongo only to see it carom into the Vancouver net to tie the score at 9:07. The goal was Vermette's second of the night and 17th of the season.
The Canucks had just 10 shots in total when Higgins restored Vancouver's lead at even strength by deflecting a shot from the point past Smith for his 14th of the season and second in as many games at 16:59.
Dalpe then followed up his own rebound off the rush to score his third of the season and stretch the Canucks' lead to 4-2 just 42 seconds later after a great play by Vancouver's Zack Kassian at the Coyotes' blue-line.
That two-goal lead would be short-lived as Vermette completed his second hat trick of the season — and third of his career — with 54.5 seconds remaining in the period with a shot from in close as the Coyotes broke through against the Canucks' stingy penalty kill, which came in ranked second overall. Vermette now has six goals in his last six games, and three short-handed goals in his last four.
The final three goals of the second period came in a span of 2:06.
The Canucks — who were still without injured forwards Henrik Sedin and Mike Santorelli because of upper-body injuries — came into Sunday having scored just eight times in their last seven games, but beat Smith twice in a 44-second span midway through first period as both Bieksa and Hansen ended lengthy droughts.
Bieksa opened the scoring at 10:31 on a shot through traffic to pick up his first goal in nine games. The defenceman's third of the season was also his first point in the last eight contests and came moments after a failed Vancouver power play.
Hansen then doubled the Canucks' lead at 11:15, ripping a shot five-hole on a 3-on-2 rush for his ninth of the season to snap an 11-game goalless streak. The breakthrough also marked the Danish forward's first point in nine games.
The fans at Rogers Arena were barely back in the seats when Phoenix cut the lead in half just 18 seconds later thanks to a gaffe by Luongo, who inadvertently directed a loose puck off the stick of Vermette into his own net.
Luongo made up for that mistake later in the period, stopping Coyotes forward Mikael Boedker twice from in tight.
The Canucks' power play came into Sunday on a dreadful 2-for-30 run over the last eight games.
Notes: Henrik Sedin missed his third straight game, while Santorelli sat out for the fourth consecutive game. ... Canucks forward David Booth sat out as a healthy scratch with Vancouver electing to dress seven defenceman. Yannick Weber took some shifts as a forward, but played primarily on the first power play unit. ... Sunday's matchup was the fourth of five meetings between Phoenix beat Vancouver this season. The Coyotes scored a 3-2 shootout victory on home ice in November, before the Canucks won 3-2 in overtime at Rogers Arena in December. Phoenix then beat Vancouver 1-0 in the desert on Jan. 16. The teams close out their season series on March 4 in Glendale, Ariz. ... The Canucks play the third game of their four-game homestand on Monday against the Edmonton Oilers. The Coyotes host the Los Angeles Kings on Tuesday in the opener of a five-game homestand.