Luongo made 43 saves - 19 on power plays - and Cowan scored 7:49 into the second overtime Friday night to lift Vancouver to a 2-1 victory over the Anaheim Ducks.
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 goalie 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.
"Trevor's a clutch player, and he's proved it again and again in these situations," Cowan said. "It's exciting to be on the ice with him at that time of the game. He got the puck to me at the right time and I just got it away quick.
"For a second, I thought it was in, and then I was like, 'Aw, maybe it's not,' and then all of a sudden I got bear-hugged from behind. It was very exciting. It's my first (overtime goal in the playoffs), and I'll always remember it."
It was the first post-season goal Giguere has ever allowed in overtime, an NHL record run of 197 minutes, 51 seconds of OT. The Anaheim goalie finished with 47 saves.
"Luongo had some big saves and we had some opportunities that were close," the Ducks' Scott Niedermayer said.
Luongo said the Vancouver defence played well in front of him.
"After Game 1, it was important to make sure there were no second opportunities in front of the net, and tonight those guys stepped up and did a wonderful job for me," Luongo said, referring back to the Canucks' 5-1 loss in the series opener.
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 and Canucks 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 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.
"They were settled into the series a little bit and played better," Anaheim captain Scott Niedermayer said. "We came out a little slower than we wanted to. We can be better, for sure."
Luongo said the Canucks remained confident despite the lopsided loss in the first game.
"We knew we could get the job done," he said. "It was our fourth overtime game of the playoffs, so guys were aware of what needed to be done to get the win.
"We had to stay patient and try not to make any mistakes."
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 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.
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.