Michael Ryder scored two goals and added an assist to lift the Stars over the Oilers 3-1 on Wednesday night.
Mike Ribeiro also scored and had two assists for the Stars (42-30-5), who moved back atop the tight Pacific Division. Dallas has a one-point lead over Los Angeles and San Jose, and a two-point edge over Phoenix. Each team has five games remaining.
The Stars jumped from seventh place to third in the Western Conference standings.
"Every point is so valuable the way things sit right now," Dallas defenseman Sheldon Souray said. "We have a tough road right to the end of the season, and hopefully we will be in the mix when the dust settles a week or so from now. It's cliche, but you can only take it one game at a time.
"We've been in third for a while this year, and you take one loss and you are in seventh or eighth and battling with teams one point behind you. It's exciting with so much on the line right now."
Kari Lehtonen was sharp for the Stars with 31 saves.
"It was a big two points for us," he said. "Sometimes it is tougher to play against the teams that are already out because they might be more offensive-minded than the ones in the playoff race. Every game is tough right now. We are happy to get these two points and keep going."
Ryder moved past the 60-point mark for the first time since his rookie season with Montreal in 2003-04.
"I'm just grateful for the opportunity here and for the chance to play on the top line," he said. "We've been working well together. I'm happy to come here and help this team make the playoffs, and we are right there now."
Ales Hemsky had the lone goal for the Oilers (31-37-9), who were 5-0-2 in their previous seven games.
"We kind of fell into their trap," Oilers coach Tom Renney said. "We weren't very good in our own end or defensively, which includes the forecheck.
"We needed to stick to the job and get out noses a little dirtier than we did."
Dallas swept the four-game season series from the Oilers, who are ahead of only Columbus in the West. When asked whether the Stars were the one team that had Edmonton's number this season, Renney responded with a laugh.
"We only have one?" he said. "That's awesome. We are improving."
Edmonton came close to going ahead early, just eight minutes in, on a short-handed 2-on-1 break, but Lehtonen came across to make a kick pad save on Nick Schultz.
The Stars took a two-goal lead 5:05 into the second period. Ribeiro took a return pass from Loui Eriksson on a 2-on-1 rush and banked the puck into the net off Dubnyk.
Ryder had a breakaway chance, but Dubnyk made a stick save to keep it at 2-0 after 40 minutes.
Lehtonen kept the Oilers off the board early in the third, doing the splits to stop Jordan Eberle on a short-handed break.
The Stars made it 3-0 at 4:11 of the third when Ryder finished a tic-tac-toe passing play at the side of the net for his second goal of the game.
Edmonton broke Lehtonen's bid for his fifth shutout of the season at 8:21. Corey Potter's shot went wide but came off the boards to Hemsky at the other side of the net, and he scored into a wide-open goal.
The Stars won the first three games of the season series by a combined score of 11-3. It was their first sweep of the Oilers since the 2007-08 season. ... Oilers captain Shawn Horcoff was in the lineup after leaving the previous game early with an injury. Rookie F Lennart Petrell and veteran D Andy Sutton also returned. ... F Reilly Smith played for Dallas but the Stars were without forward Eric Nystrom and Radek Dvorak, and defenseman Mark Fistric. ... Eriksson had two assists to reach 70 points for the third consecutive season. Steven Stamkos, Joe Thornton and Henrik Sedin are the only other NHL players to have done that.