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 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.
The goal ended an NHL record shutout streak by Giguere. The goal was the first he has ever allowed in a post-season overtime game, with Cowan's score finishing Giguere's run at 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 match.
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.