Nazem Kadri had a goal and an assist for the Flames (21-18-5), who have won four in a row for the first time this season. Vladar made 23 saves.
"We want to put on a good show and we're fighting to make the dance, so every single game is magnified for us," Kadri said. "We're trying to show that desperation early, and we're coming together, rallying as a team, and that's a great sign."
Sean Durzi had a goal and an assist, and Ingram made 27 saves for the Coyotes (21-18-3), who have lost five of seven.
"Points are so important now this time of year, especially this next little stretch," Durzi said. "You'll see it coming into the (All-Star) break. We know where they are. We know where we are. We know that two points are everything. Obviously, it's disappointing, but this time of year we have to be able to, at least from my experience, move on. That's it. What's happened has happened."
Coleman cut the lead to 2-1 at 8:54 of the third period, sliding a shot past the right pad of Ingram from the edge of the crease.
"It's a team we're fighting with for points to climb into that wild card picture," said Coleman, who has eight points (five goals, three assists) during a four-game point streak. "We just couldn't let it slip away no matter how we felt. That competitive spirit comes out of you and you find ways to dig deep and try to get on the board."
Kadri tied it 2-2 at 9:57 when he converted a one-handed cross-crease pass from Martin Pospisil.
"Great play," Kadri said. "His speed's one of his best attributes. He was able to get that out front to me, and obviously it was a huge goal. Great play by him. It's an emotional goal, a big goal for us. It allowed us to climb back into the hockey game. It allowed us to bounce back."