The Canucks clinched the Western Conference title with the Game Five victory and will face either the Boston Bruins or the Tampa Bay Lightning in the Stanley Cup finals.
Bieksa slammed in the winning shot after the puck came to him via a strange bounce off the glass that seemed to fool everyone. Sharks goalie Antti Niemi never saw the puck until it was too late.
"I think I was the only one on the ice who knew where the puck was," Bieksa said. "You know it is going to be a good shot when the goalie is looking the wrong way.”
Alexandre Burrows opened the scoring for Vancouver in the first period before Patrick Marleau equalised with a power-play marker in the second period and Devin Setoguchi put the Sharks 2-1 up just 24 seconds into the third.
But just when it looked like the Sharks would send the series to a Game Six, Ryan Kesler forced overtime by scoring with 14 seconds left in regulation.
The Canucks, who are celebrating their 40th year in the National Hockey League, have never won the league's ultimate prize but finished the regular season with the best record.
"This has been a long time in coming," said Vancouver coach Alain Vigneault, who credited the team's depth of talent including goalie Roberto Luongo.
Luongo, who helped lead Canada to Olympic Gold at the 2010 Vancouver Winter Games, was in inspirational form on Tuesday, making several spectacular overtime saves among his 54 stops.
"I was having fun out here. I was in the zone," Luongo said.
It marked the second consecutive season that San Jose reached the conference finals only to be denied a place in the Stanley Cup finals.
Coach Todd McLellan still felt the Sharks were an elite team and said it was especially tough to get knocked out of the playoffs on such an unusual bounce of the puck.
"Our task ahead of us is to get our asses back here next season," McLellan said.