Schroeder, who hasn't scored a traditional goal in the regular season, fired a low, hard shot that squeaked through the pads of goalie Corey Crawford.
Alex Edler scored in regulation for the Canucks (4-2-2).
Patrick Kane had the lone goal for the Blackhawks, who lost their second straight in a shootout after starting the season with six victories.
Chicago controlled overtime, outshooting Vancouver 4-1. The Blackhawks were also awarded a power play with 1:11 left when Vancouver defenseman Jason Garrison was called for high-sticking. Chicago, which outshot Vancouver 28-22, was 0 for 5 on the power play.
The Canucks had a 40-second, 5-on-3 power play in the third period, but didn't register a shot on goal.
Edler opened the scoring at 15:42 of the first period. The big defenseman took a pass from Zack Kassian and fired a shot past Crawford for his third goal of the season.
Kane tied it at 9:42 of the third. He got the puck from Andrew Shaw and lifted a shot over the stick side of goalie Roberto Luongo, who was falling to the ice. It was Kane's third goal.
Vancouver's Mason Raymond had a great scoring chance late in regulation. His hard shot went through Crawford's pads and slid across the open net.
The Blackhawks used speed and some pretty passing to attack Vancouver. Despite being drawn out of position at times, the Canucks defense forced the Blackhawks to the outside and blocked plenty of shots.
Luongo, who stopped 24 shots in a win against the Colorado Avalanche on Wednesday, had another solid night. In the second period he gloved a drive from Brandon Saad and calmly blocked a long shot from Dave Bolland.
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In the first period, Luongo stopped Marian Hossa on a breakaway and then dragged his leg in front of Brent Seabrook's drive during a Chicago power play.
Luongo also got help when a couple of Blackhawks drives struck goal posts.
Luongo, the goalie the Canucks have been trying to trade since June, made his third consecutive start. Cory Schneider, who was expected to be Vancouver's No. 1 goalie this season, watched from the bench.
It was also the first meeting between the teams since March 21 when Chicago defenseman Duncan Keith delivered an elbow to Daniel Sedin's head.
Sedin sustained a concussion and missed the rest of the regular season plus the first three games of the playoffs. Keith was given a five-game suspension.
The crowd booed Keith every time he touched the puck. There was a cheer in the second period when Henrik Sedin bounced Keith into the boards.
NOTES: The Canucks wore the Vancouver Millionaires "V" patch on their sweaters to celebrate the team that won the Stanley Cup in 1915 against the Ottawa Senators of the National Hockey Association. ... Wes Rypien, brother of former Canucks player Rick Rypien who committed suicide, participated in a ceremonial opening faceoff. ... Henrik Sedin played his 900th game with the Canucks. ... Canucks grinder Dale Weise played despite missing practice Thursday with a stiff neck. Weise fought Cody McLeod during Vancouver's 3-0 win Wednesday.
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