The goal came 2:23 after a Rittich turnover behind the Calgary net allowed Hayes to score with 3:46 left to give the Flyers a 2-1 lead.
Rittich's teammates refused to let their goalie be responsible for another loss.
"[Rittich and goalie Cam Talbot have] been playing great our whole season so far," Mangiapane said. "They've been bailing us out, we've got to bail them out, it's time for us to help [Rittich] out there. We knew we had to bail him out. That was a message on the bench, we weren't letting up here and we're playing good. Let's keep going, let's get one on the 6-on-5."
Mangiapane's goal, at 1:01 of the third, tied it 1-1. It also ended Calgary's road shutout streak at 261:03, eighth-longest in NHL history and the longest since the Phoenix Coyotes went 296:51 from Feb. 26-March 19, 2013.