CHICAGO -- San Jose coach Todd McLellan gushed about the Sharks' defense and effort. He also praised goalie Alex Stalock.
What a difference a day makes.
"This was the best defensive effort we've had in this building in a while," McLellan said.
Stalock made 33 saves through overtime as San Jose bounced back from a 4-3 loss at Colorado on Saturday. Jason Demers scored in the first period, and Brett Burns had a tying goal in the third.
The Sharks allowed the first four goals in the loss to the Avalanche, and McLellan pulled Antti Niemi early in the second period in favor of Stalock. McLellan then talked about holding the goalies accountable in 2014, along with the rest of the team.
"It's hard to demand perfection, but you have to demand as close as you can get to that," McLellan said. "The players tonight gave us that. It still was a tough task. To score only two and win a game against Chicago, you've done something pretty well."
Couture beat Corey Crawford with a shot into the right side of the net after Stalock stopped Chicago captain Jonathan Toews leading off the shootout. Patrick Sharp then shot wide before Pavelski scored to give the Sharks their first win at the United Center since Dec. 30, 2010.
Crawford made 40 saves in his first home start since he left Chicago's 6-2 victory over Florida on Dec. 8 because of a lower-body injury. He came back on Thursday and stopped 31 shots in a 3-2 overtime loss at the New York Islanders.
"It's a tough loss to take when you're down by a goal, get up by one, and they tie it up again," Crawford said. "The shootout wasn't close, obviously. Tough loss."
Defensemen Niklas Hjalmarsson and Michal Rozsival scored in the third period for the Blackhawks (29-7-9), who have earned at least one point in their last 10 games. It was their fifth home shootout loss.
"We've lost a few in a row here and I think it could be a little bit uncomfortable, or a lack of confidence, but we've got to work our way through it," coach Joel Quenneville said of the shootout trouble. "We've left a lot of points on the board.
"We could have put ourself in a better spot than we're at. That's been a sour spot recently."
San Jose (27-10-6) trailed 2-1 before Joe Thornton made a great pass across the crease to Burns, who slammed the puck in at 11:10 of the third. Thornton has 12 assists during an eight-game point streak, running his NHL-best total to 43 on the season.
San Jose had dropped six straight in Chicago by a combined score of 24-11, including a 5-1 loss on Nov. 17. The Sharks and Blackhawks are two of the top teams in the Western Conference.
"Any win in here is special," Thornton said. "It's a hard place to play. They play fast. They play hard, and any win in this building is huge."
The Blackhawks had few quality scoring chances in the first two periods, with most of their shots coming from the outside. Stalock grabbed most of them cleanly, and San Jose swept away any potential problems around the net.
The NHL's highest-scoring team appeared to be headed for its first shutout of the season before two unlikely players combined to put the Blackhawks in front.
First, Hjalmarsson skated toward the middle of the ice, got a pass from Toews and beat Stalock at 1:27 of the third. Then Rozsival's long slap shot through traffic made it 2-1 just 2:34 after Hjalmarsson's fourth goal.
It was Rozsival's first since March 1, 2012, for Phoenix. Hjalmarsson hadn't scored since Nov. 30 at Phoenix.
The Sharks grabbed the lead midway through the first period. They had an extended possession in the Blackhawks' zone before Demers skated into an open area on the right side of the net and one-timed a pass from Couture past Crawford.
It was Demers' third goal of the season, all in his past eight games. Couture extended his point streak to five games after scoring in each of his previous four.
Sharks forward Eriah Hayes made his first NHL appearance. He was recalled from Worcester of the AHL on Saturday. ... The announced crowd of 21,599 was the 250th consecutive home sellout for the Blackhawks, counting the postseason. ... The Blackhawks scratched defensemen Michael Kostka and Sheldon Brookbank. ... The Sharks scratched defenseman Scott Hannan and center Tyler Kennedy.