The best-of-7 series is tied 3-3 with Game 7 on Wednesday in New York (7 p.m. ET; NBCSN, RDS, TSN2, CSN-PH, MSG).
Claude Giroux had a goal and two assists and Erik Gustafsson also had a goal for the Flyers. Philadelphia also killed off all five Rangers power plays; they've stopped the Rangers now on 20 straight advantages, dating to Game 2 of the series.
Carl Hagelin and Mats Zuccarello scored third-period goals for the Rangers and goalie Henrik Lundqvist allowed four goals on 23 shots in two periods. Cam Talbot made his Stanley Cup Playoff debut in the third period in relief, stopping all five ur shots he faced.
Simmonds capped his hat trick at 15:19 of the second period, giving the Flyers a 4-0 lead. With Derek Dorsett off for tripping, the Flyers worked the puck around the New York zone. Giroux sent a pass across the zone to Jakub Voracek on the right side. He wheeled around the zone and fired a shot that went off Simmonds, who was stationed on the post on the left side, and past Lundqvist.
It was the first playoff hat trick by a Flyers player since Giroux and Sean Couturier had hat tricks in Game 2 of the Flyers' 2012 first-round series against the Pittsburgh Penguins.
Simmonds also is the first player to score a hat trick in a playoff game against the Rangers since John Madden of the New Jersey Devils did it against Kevin Weekes in Game 2 of the 2006 Eastern Conference Quarterfinals.
Gustafsson pushed the Flyers' lead to 3-0 at 14:17 of the second. In his first game of this year's playoffs, he atoned for committing a high-sticking penalty two minutes earlier by jumping out of the penalty box, grabbing a long pass from Braydon Coburn, breaking in alone on Lundqvist and firing a quick wrist shot between his pads.
"It was a lot of fun when I saw the puck come down to me," Gustafsson said. "I think it took a fortunate bounce. I almost thought Lundqvist was going to get it but luckily the puck went in."
Simmonds made it 2-0 1:32 into the second period after some hard work by linemate Brayden Schenn. Schenn took the puck away from Rangers defenseman Dan Girardi in the New York zone and broke carried it down the right side of the ice. As he cut through the slot but the puck rolled off his stick but it went right to Simmonds, who slammed it past Lundqvist.
The Rangers finally got on the board when Hagelin scored his second of the postseason at 13:26 of the third period. Anton Stralman fired a shot from high on the right side of the Philadelphia zone. The shot hit Hagelin in the right circle and dropped at his feet; he fired a quick shot that beat Mason to the short side.
The Rangers thought they had a goal 4:29 into the third when Derek Dorsett drove to the net and shoveled a shot from in close that Mason stopped. The puck popped into the air and fell behind Mason in a scrum of bodies but the referees immediately waved the goal off. Video review upheld the call when it was showed the Rangers' Brian Boyle knocked the puck into the net with his glove
Mason made a number of brilliant saves in the third. His best series came with 10:33 left when he aggressively challenged John Moore from left side and shouldered away a hard chance, then jumped over to the post on the other side to deny Benoit Pouliot on the doorstep. He also flashed a quick glove to deny Hagelin on a shorthanded breakaway with 9:27 remaining.
Giroux scored an empty-net goal with 4:11 remaining. Zuccarello closed the scoring with 56.5 seconds remaining when he beat Mason from in close off a nice set-up by Benoit Pouliot.
The Flyers took a 1-0 lead on Simmonds' power-play goal at 7:08 of the first period.
Giroux started the play when he intercepted Girardi's clearing attempt in the neutral zone and started a 3-on-2 rush with Simmonds and Scott Hartnell. Giroux found Simmonds with a cross-crease pass that Lundqvist stretched to get his right pad on. The puck made its way back to Voracek at the right point, and his shot was deflected into the left corner, where Giroux retrieved it and centered it for Hartnell in the slot. Lundqvist stopped his shot, but Simmonds grabbed the rebound and slid a shot off the skates of Rangers defenseman Ryan McDonagh, and then banged the rebound past Lundqvist.
It was the first time since Game 1 that the Flyers scored the game's first goal. It also was the Flyers' second power-play goal at home in eight chances in the series.
The Rangers had chances to even the score, including a power play when Coburn was sent off for cross-checking at 8:52. The Flyers also were credited with nine giveaways but Mason was up to every challenge, stopping all 13 shots he faced in the period.
Follow Adam Kimelman on Twitter: @NHLAdamK