Glendening made it 2-0 at 10:01 on a backhand shot.
"We started this game the way we have the last number of games. We played very, very good hockey for two periods," Red Wings coach Jeff Blashill said. "Our pressure was good, and I thought we had the better of the play for two periods. We found a way to play good and win. We talked about that. It's certainly not satisfying to play good and lose, you have to find a way to play good and win."
Draisaitl's 27th goal of the season made it 2-1 at 4:31 of the third period.
Glendening scored his second of the game at 6:28 on a rush off the left wing to make it 3-1.
"It was frustrating," Glendening said. "Our line had been out there for the goal against, and to get one back quick was nice. But the biggest thing for this team is to get a little confidence going, and two points is huge for us."