Jaromir Jagr scored a tough goal with under 2 minutes remaining to force overtime.
The Flyers improved to 2-5 in shootouts when Richards and Carter scored on Philadelphia's first two shots. Martin Biron was perfect and the Flyers won for the second time in seven games against the Rangers.
Philadelphia jumped over Boston into seventh place in the Eastern Conference with seven games left. The Rangers are sixth in the East, but closed within three points of New Jersey and Pittsburgh - the co-leaders in the Atlantic Division.
New York played most of the game without leading scorer Scott Gomez (67 points). Gomez fell early in the first period, crashed into the boards behind the net shoulder-first, and didn't return.
Danny Briere scored twice in the third period for the Flyers.
Marek Malik and Nigel Dawes scored 1:48 apart in the third to tie it at 2.
Briere's 27th goal midway through the second gave the Flyers a 2-0 lead behind Biron's solid effort. Philadelphia coach John Stevens challenged his goalies this week to step up their games and said, "We need our goalie to be our best player from this point on."
Biron was up to the challenge until New York tagged him with two goals in only 1:48 that resulted in boos.
Malik scored his second of the season on a slap shot from above the circle. Then Dawes scored his 12th goal from the high slot with 6:11 left. The puck appeared to tick off defenseman Patrick Thoresen's leg and caromed into the net.
Briere, though, was set to win it.
Briere had some challenges his first season in Philadelphia and even was booed for a spell from fans expecting more from the big ticket free-agent acquisition. The crowd erupted when he knocked in a loose puck with 3:31 left in regulation and crashed head first into the net.
Briere's smile and the celebration was short lived.
In recent weeks, the Flyers blew a lead to Florida with 3.7 seconds left in regulation and lost in overtime, squandered a 3-0 lead in the third period and lost to Toronto, and came within a blade of blowing a two-goal lead to Atlanta on Tuesday.
Jagr delivered the crucial comeback goal in this one and in most improbable fashion. He backhanded in a rebound off an impossible angle and evened the score.
The Rangers, though, couldn't score again.
Richards' breakaway goal early in the first period was enough to spoil Steve Valiquette's starting shutout streak against the Flyers. Valiquette made a combined 51 saves in two shutout victories over the Flyers and came off the bench to beat them earlier this month (17-for-18).
