Chris Tanev put Calgary up 3-2 at 13:05 of the third period when his point shot deflected off Glendening.
"They're a big, heavy team," Oettinger said. "A lot of guys make their money around the net. I obviously knew that coming into this game. Sometimes when you throw the puck at the net good stuff happens."
The goal came after Glendening tied it 2-2 at 5:29 when he scored on a rebound following up a breakaway attempt by Fredrik Karlstrom, who made his NHL debut. It was Glendening's first goal since March 2 (25 games).
Mangiapane scored into an empty net with 25 seconds remaining for the 4-2 final.
"We're not satisfied," Flames forward Mikael Backlund said. "We want to do something really big here, we know we can do it. It's an accomplishment to win the division. It's a long season, so the guys have done a great job and I'm really proud of our team."
Tkachuk gave Calgary a 1-0 lead at 10:06 of the second period with a power-play goal he swept by Oettinger from near the goal line. He is the first Flames player to reach 40 goals since Jarome Iginla scored 43 in 2010-11.
"I was probably more excited about the 40 [goals], to be completely honest," said Tkachuk, who has scored eight points (three goals, five assists) in a four-game point streak. "But just the time of the game, everything, no score, to get the first one. Just super excited. The coolest part for me was the guys coming in, they were way more excited than I was, and I didn't think that could be possible."