"We decided to play 18 minutes and we got four goals," Mantha said. "We've got good talent in here. I think if we played the right way, we could win games."
Bernier replaced Howard after the Flyers took a 5-1 lead on Patrick's goal at 1:24 of the third.
Tyler Bertuzzi, who returned after missing seven games with an upper-body injury, scored from the slot to make it 5-2 at 2:39, and Thomas Vanek finished an end-to-end rush 44 seconds later with a goal from the left circle to make it 5-3 at 3:23.
Mantha scored his first goal in 10 games when he tipped a Danny DeKeyser shot at 8:36 to make it 5-4, and then tipped a Mike Green shot past Hart to tie it 5-5 with seven seconds remaining.
Mantha said the goalie change provided a spark.
"Goalie switch always wakes us up," he said. "It tells us that we're playing terrible. Between the guys, we knew we had to play way better that what we did in the first two periods."
The Flyers trail the Carolina Hurricanes by seven points for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"We got two points," said Jakub Voracek, who scored 31 seconds into the third to give Philadelphia a 4-1 lead. "Doesn't matter how we get them. ... Should have wrapped it up earlier to save some energy but for us it's important we got those two points."
Gostisbehere scored from the left side at 4:34 of the first period to give the Flyers a 1-0 lead.
