Alex Chiasson scored on the power play at 14:41 to tie it 2-2, knocking in a loose puck in front.
Mangiapane scored with a backhand off a pass from Matthew Tkachuk to give the Flames a 3-2 lead at 19:15.
"A lot of people in Calgary were talking before that it's must-see TV," Tkachuk said. "I don't know, I guess it was a must-see hockey game for fans. It's two teams close to each other city-wise, and fans get into it. I guess it's been a rivalry for years."
Draisaitl has five goals and 11 assists during his point streak.
"In the first 10 minutes, we looked rusty coming back from the break," Oilers coach Dave Tippett said. "We were jittery with the puck, chasing the puck, and we gave up that fluky goal. I was hoping we would start a little quicker, but once we got going, I thought in the second period that we were all right. We had some coverage issues on some goals that I would like to see a better job done on."