Jonathan Toews and Johnny Oduya also scored for the Blackhawks (17-4-4), who have won three straight. Chicago has gone 11-2-1 in its past 14 games.
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After starting the trip with a 5-1 loss in Colorado, the Blackhawks have regained their form.
"We've got back to our systems and rolling all four lines," Shaw said. "Everybody is bringing the energy we need them to, and the chemistry through all four lines is pretty amazing right now. We've been playing some of our best hockey.
"We've had a couple of injuries to key people, but everybody is stepping up. That's why our game has improved."
The Blackhawks moved ahead of Anaheim and into first place in the NHL standings.
Justin Schultz had the lone goal for the Oilers (7-16-2), whose season-high, three-game winning streak was snapped.
"It was a crazy game to me a little bit," Oilers coach Dallas Eakins said. "It wasn't a 5-1 game. I thought it was probably a 3-1 game. But giving up that shorty early, where we bobbled the puck, cost us.
"Then we couldn't kill that penalty, and when you're down 2-0 to the defending Stanley Cup champions, it's usually going to be trouble right away."
"They pretty much beat us in every aspect," he said. "I felt like the energy was pretty low for having three days off. It's a disappointing game for us wanting to make a statement.
"We have to find an answer in here."
Corey Crawford made 24 saves to earn his 16th win of the season. The Blackhawks lost 9-2 and 8-4 in Edmonton during the 2011-12 season.
"We are just taking it game by game and trying not to look too far ahead," he said. "We are a very focused team right now. Playing in a building where we haven't had much success of late, it was nice to come in here and get off on the right foot and get a couple of goals early."
Chicago coach Joel Quenneville said the key was his team's early scoring output.
"We had a great start," Quenneville said. "We get that short-handed goal and we have a red-hot power play that got us off on the right foot."
Chicago took a 1-0 lead five minutes in with a short-handed goal as the Oilers' Nail Yakupov couldn't corral a bouncing puck at the blue line, allowing Toews to take off on a breakaway. The Blackhawks captain waited for goalie Devan Dubnyk to commit before scoring his 12th goal of the season.
Chicago doubled its lead three minutes later during a power play as Edmonton's defenders failed to clear a puck in the crease. Kane swooped in and scored into a wide-open net with Dubnyk down and out of the play.
The goal extended Kane's league-best point streak. He has six goals and nine assists during the 11-game span.
Chicago went up by three goals 1:28 into the second period when Oduya's shot bounced off the glove of Edmonton defenseman Corey Potter and into the net.
The rout was on when the Blackhawks made it 4-0 at 5:49 on a 3-on-1 rush. Jeremy Morin backhanded a pass to Shaw, who scored into an open net.
Dubnyk was then replaced in goal by Ilya Bryzgalov, who made his Oilers debut.
The veteran goalie, who signed with Edmonton on Nov. 8, hadn't played an NHL game since he was bought out by the Philadelphia Flyers with seven years remaining on his nine-year, $51 million contract.
The goalie change seemed to spark the Oilers, and they scored on Crawford at 8:08 of the second. Schultz skated in from the point and fired a shot from near the faceoff dot to make it 4-1.
Shaw nearly had his second goal of the night during a late third-period power play, but it was immediately waved off because the puck was clearly kicked into the net.
With two minutes remaining, however, Shaw did score again when he tipped Nick Leddy's shot past Bryzgalov for his seventh goal.
The Blackhawks will play the fifth game of their seven-game trip in Calgary on Wednesday. The trip coincides with the circus arriving in Chicago. The Oilers will start a three-game trip in Nashville on Thursday.
It was the second of three meetings between the teams this season. The Blackhawks won the first, 5-4 in Chicago on Nov. 10. ... Dubnyk was selected as the NHL's third star of the week. He allowed only one goal in a pair of wins over Columbus and Florida. ... Chicago forward Marian Hossa missed the game to attend to a family matter. Fellow forward Bryan Bickell sat out his third straight game because of a lower-body injury. Blackhawks center Michal Handzus played his first game since Oct. 24, returning from a hand injury.
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