Dylan Larkin had a goal for the Red Wings, who snapped their two-game skid. Jimmy Howard made 32 saves.
Detroit got on the board four minutes into the second period when Larkin scored his 21st of the season on a quick wrist shot.
The Oilers got caught scrambling midway through the second, leading to a backhand goal by Glendening.
Draisaitl swatted in a puck out of midair for his 27th at 4:30 of the third period as Edmonton finally scored. Detroit restored its two-goal lead when Glendening somehow squeezed in a bad-angle shot past goalie Mikko Koskinen.
The Oilers pulled Koskinen early and it paid off with 2:26 left on a point drive from Benning, but they were unable to tie it.
NOTES: The Red Wings improved to 4-9-2 in their past 15 games. … Edmonton won the only other meeting this season, 4-3 on Nov. 3. … Koskinen made his first start for the Oilers since signing a three-year contract with an average annual salary of $4.5 million on Monday. … Two players the Oilers placed on waivers Monday, Ty Rattie and Ryan Spooner, cleared waivers Tuesday. Spooner dressed against Detroit, while Rattie was a healthy scratch.
Red Wings: Off until Feb. 1, when they host Toronto.
Oilers: Play at Philadelphia on Feb. 2.
Source: Fox Sports