Lehtonen made 27 saves for his 18th career shutout, Goligoski scored in the first period, and the Stars ended Anaheim's five-game winning streak with a 1-0 victory over the Ducks on Tuesday night.
"Our best player was our goalie ... you saw a great performance by a great goaltender tonight," Stars coach Glen Gulutzan said.
The Stars had been 0-4-1 with six goals in their previous five games, damaging their playoff chances.
Dallas boosted its points total to 52 heading into the All-Star break, pulling the Stars closer to the Western Conference's top eight.
Goligoski wasn't worried about style points on his ugly goal, just results.
"It feels good to get that one there and get the lead," said Goligoski, who signed a four-year contract extension worth $18.4 million on Monday. "We haven't played with the lead much lately."
Anaheim got off to a fast start with a 9-2 edge in shots before Goligoski notched his sixth of the season.
Goligoski's shot glanced off the right hip of Ducks defenseman Cam Fowler, with the puck flipping into the air and over Anaheim goaltender Jonas Hiller's shoulder with 3:39 left in the first. Mike Ribeiro and Loui Eriksson were credited with assists.
Fowler was never able to locate the puck until it was in the net.
"It was kind of a broken play, three or four guys were in front of me," Fowler said. "I didn't even see the shot come. I don't know what I would have done differently to avoid it. It's kind of mind-blowing that it was the only goal in the game. But it's part of the game. We had our chances, too, and didn't bear down."
Hiller stopped 15 shots for the Ducks, who had been 8-0-1 in their previous nine games.
Once 20 points out of the eighth spot in the West, the Ducks have returned to the fringe of the playoff race. But they still have some catching up to do, starting the night 11 points out of the final postseason berth.
"We had our chances but we didn't capitalize," said Ducks All-Star right winger Corey Perry. "We've worked our way up and we're battling to achieve our goal. So it's a tough way to lose."
Ribeiro was back after missing seven consecutive games because of a knee injury.
Anaheim held a 19-7 shots-on-goal advantage after two periods, but the Ducks couldn't get the puck past Lehtonen, who earned his first shutout of the season.
Lehtonen got plenty of help from his teammates, who blocked 27 shots -- seven by defenseman Trevor Daley.
"They got a couple of good chances, but overall, especially in the third, we played pretty well," Lehtonen said. "They didn't get many shots and I think that was the big key. Usually, when you're up one goal, the other team comes real hard, but we were able to get the puck deep and play solid."
Lehtonen denied Perry from close range seconds before the final buzzer.
"Give (the Stars) credit, in the third period they were a pretty determined group checking-wise," Ducks coach Bruce Boudreau said. "When you haven't won in a couple weeks, you're gonna check your rear end off and they did. They frustrated us."
Dallas was able to control Anaheim's hottest scorer, right winger Bobby Ryan, who had four goals in his previous five games. Ryan was credited with one shot, and had three others blocked by the Stars.
Dallas is 60-27-13 all-time against Anaheim. ... The Ducks are without G Dan Ellis (groin) and RW Devante Smith-Pelly (foot). ... Dallas LW Matt Fraser made his NHL debut, skating on the fourth line with C Jake Dowell and RW Tomas Vincour. ... Stars LW Vernon Fiddler played in his 500th NHL game. ... Hiller had been 6-1-1 in his previous eight starts. ... Anaheim was shut out for the fourth time this season. ... The Stars didn't have a power play chance, the first time the franchise has won a game without a man-advantage opportunity since the team moved to Dallas in 1993.