Ottawa, which solidified its hold on seventh place in the East, scored on both of its shootout shots, getting goals from Milan Michalek and Jason Spezza.
Matt Read gave the Flyers a 1-0 lead in the shootout, but Giroux - who, oddly, went in backward - and Danny Briere could not convert against goalie Craig Anderson.
Giroux said the fact he was stopped on a breakaway earlier in the game is not why he changed his approach in the shootout.
"I've tried this move a couple of times, and he kind of surprised me with the poke check," Giroux said. "If he doesn't poke check, it's a goal. Obviously, I'd like to have it back."
The Flyers are 4-7 in shootouts this season.
With 2 minutes, 47 seconds left in a wildly entertaining overtime, Briere made a dazzling move to get free for a shot, but fired off the right post.
"I saw the puck going over his shoulder and thought for sure it was going in," said Briere, whose team had the better OT chances and finished with a 39-29 edge in shots. "It was one of those lucky breaks for him, and a bad bounce for me."
Flyers goalie Sergei Bobrovsky had kept the game tied at 3-3 with a left pad save on Spezza out front with 1:20 left in regulation.
"Even though he let up those three goals, he came back and slammed the door," said Simmonds, who said he had six or seven stitches administered above his right eye and the bridge of his nose, after he deflected Brayden Schenn's shot into the net. "He made some huge saves to allow us to get that one point."
During the last two weeks, the Flyers' early play has resembled developments in that old Bill Murray movie, Groundhog Day.
They fall behind early. They allow the game's second goal. They rally furiously.
Over and over and over again.
For the fifth time in the last eight games, the Flyers fell into a 2-0 hole.
This time, the deficit grew to 3-0 before they scored three unanswered goals in the second period to knot the score.
"We have a lot of character and never quit," said rookie center Sean Couturier, who ended an eight-game goalless drought by scoring the Flyers' second goal. "We believe in ourselves."
In four of the last five games in which the Flyers have faced a 2-0 deficit, they have rallied to tie the score or take the lead.
Ottawa stormed to a 3-0 first-period lead as turnovers led to all three goals.
A little over six minutes into the game, Pavel Kubina's turnover contributed to Spezza's goal. About 51/2 minutes later, Anderson, who was brilliant in the first period, stopped Claude Giroux on a shorthanded breakaway, turning aside his backhander.
Later in the period, Daniel Alfredsson scored from the high slot after an Eric Wellwood turnover, and Bobrovsky made two gaffes that led to Chris Neil's goal from deep in the left circle. Bobrovsky turned the puck over on a bad clearing pass, and then saw Neil's bad-angle shot trickle off his pads and into the net, giving Ottawa a 3-0 lead.
But the Flyers got second-period goals from Simmonds, Couturier and Braydon Coburn to tie the score at 3-all.
Simmonds, who went to the locker room to have his face stitched after his goal, returned to the ice later in the second period and received a rousing ovation when he stepped back on the ice. His goal was his 26th, one behind Giroux for second on the team.
Contact Sam Carchidi at email@example.com or on Twitter@BroadStBull.