It's the closest his team will get this year.
Jeff Skinner added a pair of goals for the Hurricanes, who finish the season in seventh place in the Metropolitan Division, outside of the playoff picture.
Wayne Simmonds scored twice for the Flyers, who had already wrapped up third place in the Metropolitan Division and a date with the New York Rangers in the first round of the playoffs later this week.
"Games like this are good for our team," Muller said. "Not that it's a playoff game or anything but this is a tough rink to play in, it was sold out and the guys were intense until the end. For our guys to find a way to win was what we like to see as coaches. It was a good finish for us."
Cal Heeter made 33 saves in his NHL debut for the Flyers, before stopping one of two shots in the shootout. Heeter was recalled from the Philadelphia Phantoms and given the start after Steve Mason suffered an upper-body injury in Saturday's win over the Penguins.
Mason is day-to-day but the Flyers are hopeful he'll be able to return to net for Game 1 of their series with the Rangers.
"He battled back," Flyers head coach Craig Berube said of Heeter. "It's tough for a kid like that coming in here. He's nervous obviously and this is probably not the best situation (for his NHL debut). But it is what it is and the kid hung in there and did a good job."
Anton Khudobin made 39 saves and stopped all three of shots in the shootout for the Carolina.
Skinner scored his 33rd goal of the season on a power play with 4:21 left, giving the Hurricanes a 5-4 lead after the puck took a weird carom off the boards and left him shooting at an open net.
But Simmonds tied the game, scoring on a rebound with 9.5 seconds left for his career-high 29th of the season.
Fighting back in the final minute felt like a positive sign to the Flyers heading into the postseason.
"If we play as a team and keep doing what we've been doing the past few months, we'll be a really tough team to beat," Flyers defender Kimmo Timonen said.
Trailing 4-1, the Flyers went on a second-period offensive, scoring three goals to tie the game, including two within 10 seconds of each other -- on a power-play slap shot from the point from Timonen and a wrister from Simmonds.
With less than five minutes remaining in the period, Sean Couturier tied the game with a steal and pretty move on a breakaway, matching his career high with his 13th goal.
"I thought we skated and were competitive the whole game," Berube said. "We were just looking for good habits and getting our skating legs, and I thought that they did that."
Heeter's debut got off to an inauspicious start as he failed to stop the first shot he faced, with Skinner beating him with a wrist shot.
The Hurricanes scored twice more before the first period ended, with Staal beating Heeter on a low shot from the left circle and Manny Malhotra going five-hole on the rookie for his first goal since Jan. 25.
Staal scored his second just 17 seconds into the second period on an easy shot from the right circle.
"In my dreams, I'd say I didn't give up six goals," Heeter said. "But it's a fabulous experience. I've been working my whole life to get to here and to finally get the opportunity to play a game and to have the support of the management, players, coaches, fans, everybody here . it's a dream come true."
The Flyers scored their first goal midway through the first period when Matt Read blasted in a one-timer from Jason Akeson, who logged his first NHL assist. Akeson, who was called up from the Phantoms for Sunday's game, has two points in two career NHL games.
Before the game, Claude Giroux was presented with the Bobby Clarke Trophy, given to the team's MVP as voted on by local sports writers and broadcasters. Mason, Timonen, Jakub Voracek and Michael Raffl also won team awards, and Vincent Lecavalier was honored for recently reaching the 400-goal milestone. . The Flyers finished 0-2-2 against the Hurricanes this season, marking the first time they went an entire season without beating the Carolina/Hartford franchise since 1991-92.