Kane scored the Blackhawks' second goal of the game on the power play, giving him six goals and nine assists during his streak.
Jonathan Toews, Johnny Oduya and Andrew Shaw (two goals) also scored for the Blackhawks, who have won three of their first four games of their seven-game road trip and have taken over the top spot in the NHL standings.
Chicago goalie Corey Crawford made 24 saves for the win, improving his record to 16-4-3 on the season.
"We are just taking it game by game and trying not to look too far ahead," Crawford said. "We are a very focused team right now. Playing in a building where 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."
Justin Schultz scored the lone goal for the Oilers, who had a three-game winning streak snapped.
Devan Dubnyk turned aside 10 shots before being pulled in the second period after the Blackhawks' fourth goal.
He was replaced by Ilya Bryzgalov, who was made his Oilers debut. Bryzgalov was signed as a free agent by Edmonton earlier this month, but had been backing up Dubnyk while the Oilers put together their win streak with victories against the Calgary Flames, Columbus Blue Jackets and Florida Panthers.
"We do a lot of work every day, and I felt pretty good," Bryzgalov said. "On the one side, we're losing 4-0 and it's not exactly what you want for the team. On another hand, I got some opportunity to get some game, make sure all system right and find out maybe where's the problems. If I get next start, at least I gotta know where am I right now."
Toews opened the scoring 5:14 into the first period on a shorthanded breakaway. The Blackhawks captain was first to the puck after Nail Yakupov misplayed it at the point. Toews went in on Dubnyk and scored with a backhand-to-forehand move.
"We had a great start in the game," Blackhawks coach Joel Quenneville said. "We get that shorthanded goal and we have a red-hot power play that got us off on the right foot. Getting off to that start was important. The special teams in the first period really put us in a good spot. Across the board, we did what we were looking to do and played a really solid game.”
Kane increased the Blackhawks' lead via the power play at 8:06, picking up a loose puck in front after a scramble.
"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."
In the second period, Oduya was credited with the Blackhawks' third goal at 1:28 after Oilers defenseman Corey Potter inadvertently swatted the puck into his own net.
Shaw made it 4-0 at 5:49, when he was set up for an easy tap-in by Jeremy Morin on a 2-on-1 rush.
Schultz scored the Oilers only goal at 8:08, taking pass from Ryan Nugent-Hopkins at the right circle and firing a shot past Crawford.
Shaw had a goal waived off midway through the third period, when he kicked the puck past Bryzgalov. He then scored at 18:02, tipping a Nick Leddy shot past the Oilers' goaltender.
"I needed to make up for a few bad plays I had in my last game, so I just tried to work extra hard tonight," Shaw said. "We've got back to our systems and rolling all four lines. 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."
Dubnyk, who was named the NHL's Third Star for his performance last week, gave way to Bryzgalov after the Blackhawks' fourth goal. Bryzgalov made 12 saves in under 35 minutes.
"That's a veteran guy. He comes in with a little swagger," Eakins said of Bryzgalov. "I thought he was good. The only one that got by him was tipped."
The Blackhawks continue their road trip Wednesday against Calgary. The Oilers visit the Nashville Predators on Thursday in the first of a three-game road trip.
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