The win evened their Western Conference semifinal series at one game apiece.
Cowan scored from a sharp angle with a shot that found a crack of daylight between goaltender Jean-Sebastien Giguere's right skate and the post after Anaheim's Rob Niedermayer lost the puck behind the net to Vancouver centre Trevor Linden.
It was the first goal Giguere he has ever allowed in a post-season overtime game, an NHL record run of 190 minutes, 50 seconds.
Giguere had 12 saves in a scoreless third period and Luongo made six. The Anaheim goalie finished with 46 saves.
The teams traded goals in the second period, with Markus Naslund scoring for Vancouver at 6:30 of the period and Travis Moen answering for Anaheim 4:31 later.
The series shifts to Vancouver for games on Sunday and Tuesday. The Ducks won Wednesday's opener of the best-of-seven series 5-1 on their home ice.
The Canucks, who didn't finish their opening series with Dallas until a Game 7 victory on Monday night, skated with considerably more energy than they had in the opening loss to the Ducks.
Anaheim had a five-day break between games after wrapping up the series against Minnesota in five games, and dominated Wednesday's opener against Vancouver.
The Canucks played three overtime games in their first-round series against Dallas, winning two. Luongo made 72 saves in Vancouver's 5-4 victory in the fourth overtime of the first game.
The Ducks were playing their first overtime of this post-season.
Moen tied it 1-1 on a rebound after Luongo stopped Teemu Selanne's shot on a breakaway. The Vancouver goalie sprawled to his left to stop Selanne's shot, but could not control the puck.
Moen, trailing, poked it past Luongo, with the goalie flat on his back as the puck dribbled across the line. The goal was Moen's second of the post-season.
Naslund had given Vancouver the lead with an unassisted goal. He intercepted defenceman Sean O'Donnell's attempted clearing pass near the blue-line along the right boards, sped around defenceman Chris Pronger and got off a shot as he fell forward in the left circle.
The puck sailed past Giguere high on the stick side to give Naslund his second goal of this year's playoffs.
Moments after Moen's goal evened it, the Ducks came close to taking the lead - but Corey Perry's wrist shot from the right circle bounced off the far post after going between Luongo's legs and the Canucks' goalie smothered the puck in the crease.
Luongo and Giguere each made several fine saves during the second period, Luongo gloving Perry's slap shot from 15 feet, and Giguere snaring Brendan Morrison's shot from similar range after a turnover in the Ducks' zone.
Notes: The Canucks again were without two of their top defencemen, Sami Salo and Kevin Bieska. Both also missed the first game, and are listed as day-to-day. ... Anaheim F Brad May, suspended for three games for punching Minnesota D Kim Johnsson in the back of the head in the fourth game of their first-round series, will be eligible to play Sunday in Vancouver.