"Good teams know how to win games in the third period, and teams that struggle are the ones that lose games in the third period," said winger Matt Read, who scored a goal in his third straight game. "We have to learn how to win those games."
If not, they will fail to make the playoffs for the second consecutive season.
The Flyers received a gift - a power play for too many men on the ice - with 2:55 to go, but couldn't get the equalizer. They have been outscored by 14-5 in the third period this year.
"I thought maybe we were past that," coach Craig Berube said of the third-period meltdowns, "but we weren't tonight. You can't be that sloppy with the puck and expect to win the game."
All three Anaheim goals were scored after Flyers turnovers.
Steve Mason made three saves - including a rebound by Patrick Maroon, a former Flyers farmhand - during an Anaheim flurry with about eight minutes left to keep the score tied at 2-2. At that point, the Flyers had been outshot, 13-3, in the third period.
The Ducks finished with a 16-6 shots advantage in the final period.
"We just didn't answer," winger Wayne Simmonds said after contributing a pair of assists. "It's really disappointing. We had the game. We're up 2-0 and we kind of just stopped playing. We stood around and were watching. We started watching them play."
Palmieri scored on a breakaway after taking the puck away from Vinny Lecavalier to tie the score at 2-2 after 1:01 of the third period. Palmieri used a backhander to put the puck through Mason's legs.
Anaheim, which had a first-period goal disallowed because it was tipped in with a high stick, had cut it to 2-1 on Andrew Cogliano's tally with 3:35 left in the second.
Flyers defenseman Nick Grossmann turned the puck over to Ryan Getzlaf in the neutral zone, and Getzlaf skated into the zone and his shot caromed crazily off the boards to Cogliano, who knocked in the rebound. Mason didn't have a chance.
Lecavalier, coming off his seventh career hat trick in Saturday's 5-2 win over the Islanders, gave the Flyers a 2-0 lead by scoring a power-play goal with 3:26 left in the opening period. Claude Giroux, skating in the left circle, threaded a perfect pass through traffic to Lecavalier, who one-timed a right-circle shot past Jonas Hiller.
The Flyers' three wins were against teams with a combined 10-18-5 record entering Tuesday, so beating Anaheim (now 10-3) would have been their most impressive victory of the young season, even if the Ducks were missing injured forwards Saku Koivu and Dustin Penner, among others.
The Flyers had 16 giveaways - 10 more than the Ducks.
"I think it started in the second," Berube of the Flyers' sloppy play.
Anaheim's Teemu Selanne, a future Hall of Famer, left the ice after getting hit in the face by an apparent high stick from Luke Schenn and losing a few teeth early in the third period. Selanne did not return. Schenn was not called for a penalty.