Klingberg scored from center ice on a 74-foot, dipping slap shot that eluded goalie Viktor Fasth, and added an assist giving him three goals and eight points in eight games since being recalled from the AHL on Nov. 10.
"When you're hot, you're hot," Stars coach Lindy Ruff said. "It was a knuckler. From the bench, you could see it waffling. He probably doesn't know who Phil Niekro is, but he threw one of those."
The Stars won their third straight for the first time this season in the fourth game of a season-high, five-game homestand.
Edmonton, last in the Western Conference, is 0-6-1 in its last seven games. The Oilers haven't led since a win over the New York Rangers on Nov. 9. The Oilers fell to 0-11-1 against Western Conference opponents as they began a three-game trip against Central Division teams.
Seguin took a pass from Klingberg, who flipped the puck from deep in his zone, as he skated across the Edmonton blue line. Seguin scored to break a 1-1 tie at 5:59 of the second period, backhanding a shot over the right pad of goalie Viktor Fasth.
"I kind of stayed on the bench for an extra second and hopped on," Seguin said. "Kling looked right at me. I got fortunate with a heads-up play."
Seguin's second goal, his 17th of the season, came on a slap shot from the deep slot, off a feed by Jamie Benn from behind the Edmonton net at 17:42 of the second.
It was Seguin's fifth multigoal game of the season, also tops in the league.
"Things have been clicking, and I've been shooting the puck a bit more than I have in previous years," said Seguin, who led Dallas last season with 37 goals after being acquired from Boston. "Kind of doing more of a shooting-first mentality than always trying to pass."
Kari Lehtonen stopped 35 shots to win his third straight.
Klingberg was fed by a backhand faceoff win by Jason Spezza on the opposite side of the ice, giving Dallas a 1-0 lead at 12:49 of the opening period.
"That was a lucky goal there. I was going to dump the puck first but kind of wanted to surprise the goalie," Klingberg said.
He is also enjoying his hot start.
"Yeah, that's fun," he said. "I try not to think about it. Of course, I'm happy."
Edmonton's goals were a short-handed tally by Boyd Gordon, his fourth of the season, with only 29.6 seconds remaining in the first period, and Taylor Hall's club leading eighth and 100th of his NHL career at 5:59 of the third.
"You could argue that he is the best left winger in the game," Oilers coach Dallas Eakins said. "I'm happy that he got his 100th goal, but I know Taylor would rather celebrate this with a win."
Edmonton threatened to tie the game multiple times in the closing minutes. With about 4:30 to play, Stars center Cody Eakin cleared a loose puck on the goal line behind Lehtonen.
With 13 seconds left, Oilers forward David Perron delivered a wrap-around that struck the crossbar.
"We're shooting ourselves in the foot and beating ourselves early," Edmonton center Jordan Eberle said. "Then we wake up and find a way to come back, but it's too late."
Fasth, making his eighth start of the season, faced 34 shots. The Oilers entered the game with the NHL's worst save percentage (.883).
After a 7-1 home loss to Chicago on Saturday, the Oilers replaced goalie coach Frederic Chabot with Dustin Schwartz.
Klingberg's long goal stood out in this one.
"It took a really weird way to the net," Fasth said, "but I still got to have that. It's one of those I want back."
The Oilers hadn't scored a first-period goal since Nov. 9. ... The Stars are 11-1-2 in their last 14 games against the Oilers.