They eventually cashed in and moved into first place in the Northwest Division.
Trevor Linden and Brendan Morrison scored in the shootout, and Luongo made 34 saves to lead the Canucks to their third straight victory, 2-1 over the Chicago Blackhawks. Vancouver is 9-2-3 in its last 14 games.
The Canucks' Matt Cooke scored midway through the third to tie it one-all and set up overtime.
Had it not been for Luongo's strong play - and several close-in misses by the Blackhawks - Chicago might have put the game away earlier.
"They really made it tough on us," Vancouver coach Alain Vigneault said. "Their forecheck was tough to beat."
"Roberto gave us a chance. We were able to tie the game and win it in the shootout. He's one of the top three goaltenders in the NHL right now. He and (Martin) Brodeur are probably fighting it out for the top spot."
The Canucks were outshot 35-29 and Chicago had plenty of prime chances before Patrick Sharp opened the scoring early in the third.
Luongo's saves included one on Martin Havlat on a breakaway in the final second of overtime and a sprawling, out-of-position glove save on Denis Arkhipov early in the first.
"It wasn't our best effort, but we got two points out of it," Luongo said. "It's a good sign when we can play that way and still get the two points."
Blackhawks goalie Patrick Lalime stopped 28 shots.
Lalime, who missed the first half of the season following surgery to repair a herniated disc, made just his second start and appearance. He shut out the Canucks 3-0 in his first start, at Vancouver on Feb. 7.
"Call Luongo the best goaltender in the league," Blackhawks coach Denis Savard said. "We played well."
"Lalime played well. I won't hesitate to use him again."
Chicago outshot Vancouver 11-10 in the first. Luongo made several tough stops, including his glove save on Arkhipov 3:40 in and a point-blank pad save on Sharp in the final second of the period.
The Blackhawks misfired - or failed to get off shots - on several close-in chances early in the second.
Lalime made a tough glove save on Daniel Sedin at 7:39 of the period. He stopped a deflection by Sedin during a Canucks power play just over two minutes later.
Sharp scored at 5:13 of the third on a play following a faceoff in the left circle in the Vancouver zone.
Chicago's Martin St. Pierre fired a shot directly off the draw. Sharp pounced on the rebound in the slot and beat Luongo from 15 feet.
Havlat fired high and missed a half-open net nine minutes into the third.
Cooke tied it one-all with 9:33 left in the third when his low shot from the top of left circle was screened by Chicago defenceman Brent Seabrook and beat Lalime on the short side.
"That was just a bad goal," Lalime said.
Cooke and Kevin Bieksa failed to convert a two-on-one break 2:15 into overtime.
Linden and Morrison, the Canucks' first two shooters in the shootout, both beat Lalime. Havlat and St. Pierre failed against Luongo.
"We grinded it out," Linden said. "We've been battling to get our game together."
Notes: Vancouver C Jan Bulis sat out his third game with a shoulder injury. ... Canucks C Tommi Santala and Marc Chouinard were healthy scratches. ... Chicago D Jassen Cullimore was a healthy scratch for the fifth straight game. LW Tony Salmelainen also was a healthy scratch. ... Havlat's seven-game point-scoring streak ended. He had seven goals and five assists during the span.