In franchise history, the Flyers are 9-6 in Game 7s, including wins in their last three showdowns - at Washington in 2008, at Boston in 2010, and against visiting Buffalo in 2011.
The Rangers, who have a 5-0 franchise record in Game 7s at Madison Square Garden, have lost an NHL-record 12 straight playoff games after taking a series lead.
Wednesday's survivor will face Pittsburgh in the Metropolitan Division finals.
Simmonds scored two of his three goals on the power play, and Erik Gustafsson added a breakaway score. Claude Giroux (three points) finished the Flyers' scoring with an empty-net goal.
"Obviously, it's do or die, and I think everyone in the dressing room did a good job getting prepared for the game," said Simmonds, whose only other hat trick was in the regular season last year against Carolina. "We're not ready to go home yet. We want to move on to the second round."
The Flyers built a 4-0 lead after two periods and coasted to the win.
"I thought we played a good game tonight; we just have to carry the momentum into [Wednesday]," Simmonds said.
The Rangers see it differently.
"We're in a great position on home ice to win a series and this is over with," New York center Brad Richards said.
Just as he did in a 2-1 win in Game 4, Mason had a showstopping performance.
When he stopped Carl Hagelin on a breakaway midway through the third period, he stayed down on the ice for a few seconds, snow-angel style, leery to move because he didn't want the puck to slide into the net.
"Ma-son ... Ma-son," the fans chanted, giving the 25-year-old goaltender a standing ovation.
"Those are moments that send chills down your spine," Mason said.
"He made some huge saves for us, and he allowed us to get a couple-goal lead and then he sealed it for us," Simmonds said. "He's done it time after time this year."
Before the game, Berube said the Flyers hadn't "initiated enough in the first period yet, and we need to. Our mind-set is to come out hard early."
Simmonds apparently got the memo.
For just the second time in the series, the Flyers scored the game's first goal, a power-play tally by Simmonds.
With Simmonds stationed in front of the net, his first shot was kicked aside by defenseman Ryan McDonagh. The puck squirted back to Simmonds, and he knocked in the rebound 7 minutes, 8 seconds into the game.
Joy-struck, Simmonds leaped into the arms of Jake Voracek. In the regular season, he was third in the NHL with 15 power-play goals.
"I guess that's becoming kind of my forte," Simmonds said.
Mason made several difficult stops in the first period, including point-blank saves on Rick Nash and Anton Stralman. Mason took a .935 series save percentage into the game, and he stopped all 13 first-period shots.
"It allowed us to get our footing under us," Simmonds said.
Simmonds made it 2-0 1:32 into the second period, scoring after Brayden Schenn flubbed a shot but the puck took a fortuitous bounce to the right winger in front of the net.
About 13 minutes later, Gustafsson increased the cushion to 3-0, charging out of the penalty box, taking a feed from Braydon Coburn, and beating Henrik Lundqvist on a breakaway.
The rout was underway when Simmonds tipped in Voracek's power-play shot about a minute after Gustaffson's goal, giving the Flyers a 4-0 lead with 4:41 to go in the second.
"Obviously, we picked a bad time to have a bad period," Rangers coach Alain Vigneault said.
New York snapped Mason's shutout when Hagelin scored on a rebound with 6:34 left in the game. By then, the outcome wasn't in doubt.
And by then, the teams knew this series would have a grueling finish - games on back-to-back nights.