The rookie right wing, recalled from the minors earlier in the day, gave the Stars a 2-0 lead in his first career game.
"Coming here, first and foremost, was a great birthday present," Sawada said. "Getting that goal and the win was even better."
The Stars, who have won 10 of 13, were playing their first game since losing No. 2 scorer Richards to a fractured wrist. Richards, who had been among the team's best players after starting the season slowly, will be sidelined for as long as eight weeks after sustaining the injury Monday night in a shootout victory at Columbus.
Brunnstrom had his fifth game-winning goal of the season, which leads all rookies, after Edmonton rallied to tie it early in the second. Krys Barch, known for his physical play, had his fourth goal of the season to pick up the Stars in Richards' absence.
"Everyone knows that we have big names out of the lineup, big pieces of the puzzle," Stars center Mike Ribeiro said. "I think guys were ready for Edmonton. To lose Brad is big, but hopefully it will keep us together and keep us pushing forward."
At 1:51 of the second, Ethan Moreau was called for roughing and tripping penalties to give the Stars a four-minute power play.
The Stars capitalized just over a minute later when Brunnstrom took a pass from Loui Eriksson and beat Roloson from the right circle to give the Stars a 3-2 lead.
"We had a little bit of momentum in the second period when we came back and tied the game," Edmonton coach Craig MacTavish said. "But we got undisciplined in the penalties that we took, and that took the momentum away from us."
Dallas, seventh in the West with 65 points, won the opener of a six-game homestand that lasts until March 1.
Sawada had five goals in 48 games for Manitoba of the American Hockey League this season before his promotion to help fill-in for Richards.
After Barch scored to put the Stars on top 1-0, Sawada flipped a backhanded shot that got past Roloson at 16:36 of the first.
"Ray is a great example of how young players develop in the minors," Dallas coach Dave Tippett said. "He has been well-coached and this is a hard step. He was very good tonight."
The Oilers responded with their first goal 24 seconds later when Reddox tipped in a rebound. Edmonton tied it 47 seconds into the second on Horcoff's goal.
Edmonton is eighth in the West, three points behind Dallas.
The Oilers capped a four-game road trip with a 2-2 record. MacTavish was disappointed his team didn't play better before heading home for four consecutive games.
"The team that puts together a .750 road trip is one of the teams that's going to get into the playoffs," MacTavish said. "We could have really helped ourselves."
After Brunnstrom's go-ahead goal, the Stars built on their lead when Lehtinen scored at 8:17 of the third.
Turco improved to 17-6 lifetime against Edmonton. ... The Oilers were without RW Fernando Pisani (ankle), D Lubomir Visnovsky (shoulder) and D Denis Grebeshkov (ankle). ... Edmonton has not won a season series against the Stars since they moved to Dallas in 1993. This was the final matchup between the teams this season and they split four games. ... Five players from both teams were called for 51 penalty minutes, including three 10-minute misconducts, during one incident near the Edmonton net with 5:07 left in the game.