''Two bad plays, one mistake and then a lucky shot,'' Benn said of his second-period goal from the high slot.
It gave the Stars a 2-0 lead, and they held on to end a season-high four-game losing streak.
Benn has three goals and two assists in his last five games after going without a point for the first four games in 2016. Only Chicago's Patrick Kane and Washington's Alex Ovechkin have more goals this season.
Dallas coach Lindy Ruff had called on his team's leaders to help end the slump. Benn obliged.
In addition to his goal, the Stars captain had a game-high six shots on goal, won seven of 10 faceoffs and helped kill a third-period penalty. Oilers goalie Cam Talbot stopped Benn's attempt to deflect a shot into the net on the power play.
''Better,'' Ruff said of his top offensive players. ''That last power play, two or three real good looks to get on the board. I thought they did a pretty good job.''
Jason Spezza scored his 16th goal 1:23 into the game to give the Stars a lead they didn't lose. He also assisted on Jyrki Jokipakka's second-period goal 15 seconds after Edmonton's Teddy Purcell scored on the power play.
''(Benn)'s such a big, strong man and so calm in and around the puck that he gathered it in after it bounced three or four times and snapped a shot by us,'' Edmonton coach Todd McLellan said. ''Spezza did the same thing real early in the game, first shot on goal.''
Spezza took a pass from Johnny Oduya in the slot, skated to his left and backhanded the puck under Talbot off his stick.
Niemi finished with 23 saves. Talbot had 32.
Edmonton had scored eight goals in splitting the first two games of their three-game trip, but managed only 25 shots on goal against Dallas.
After returning from an 0-2-1 road trip, the Stars improved to 18-5-0 at home this season. Dallas (30-13-5) pulled within three points of first-place Chicago in the Central Division.
Dallas appeared to take a two-goal lead at 9:31 of the second when Jokipakka shot the puck under the crossbar. McLellan challenged the goal, and a video review showed that the Stars' Ales Hemsky was offside before Jokipakka crossed the blue line.
The lead did increase to 2-0 at 13:16. Benn recovered a loose puck and put a wrist shot from high in the slot just inside the right goal post.
Edmonton cut the deficit to 2-1 at 17:04. Purcell sent one shot off the left goal post, but regained the puck in the high slot and put a wrist shot just inside that post. The Stars have allowed five goals in 10 opponents' power plays over the last four games, but killed two of three on Thursday.
NOTES: Spezza has four goals in three games against Edmonton this season, including his first hat trick with the Stars on Oct. 13. He has 22 points (7 goals, 15 assists) in 20 career games vs. the Oilers. ... Dallas has won 10 of its last 12 home games - but the Stars are 0-5-3 in their last eight on the road. ... The Stars are 13-1-2 in their last 16 home games against the Oilers. ... Talbot has allowed only 19 goals in his last 11 games. ... Edmonton's penalty killers have an 89 percent success rate (25 for 28) in the last 10 games. In Dallas' 11 most recent games, the Stars are at 11 percent (3 for 28) on their power plays.