Meszaros ended OT with a wrist shot just outside the slot that beat Semyon Varlamov.
The winner came after the Flyers blew a 2-0 lead in the third. Jeff Carter and Claude Giroux each scored for the Flyers. Mike Knuble and Alex Ovechkin scored in the third period to rally the Capitals.
The Flyers won only hours after they gave general manager Paul Holmgren a three-year contract extension. Holmgren has made one shrewd move after another the last five years to help put the Flyers atop the Eastern Conference.
The road-weary Flyers had played only one game in 28 days and that was on Jan. 8. The Flyers even practiced at the outdoor rink at Central Park a day before they played the Rangers.
They returned home with 63 points and the top spot in the East. They can build on that lead at home - Tuesday marked the start of eight of 10 games in Philadelphia, including four of five that lead into the All-Star break.
Flyers coach Peter Laviolette is a co-coach of one of the All-Star teams - and Giroux is joining him.
The Flyers showed why they're the best in the East behind Sergei Bobrovsky in net. He stopped 12 shots over the first two periods and was in control against Ovechkin and the rest of the Caps.
Bobrovsky was cruising toward his first career shutout. He made two sweet stops on Jason Chimera - a glove save of a 2-on-1 play in the third a highlight.
All it took was 40 seconds for the lead to collapse.
Knuble, a former Flyers playoff star, scored his 11th goal to make it 2-1. Ovechkin quickly followed with his 16th goal, burning an out-of-position Bobrovsky for an easy look into the net. The play was reviewed to see if Ovechkin knocked the puck in with his glove, but the goal stood.
Lead wasted. Just not the game.
Meszros scored only his second goal of the season to stun the Caps and keep the Flyers rolling.
They all rewarded Holmgren - the architect of a team that fell two wins shy of winning the Stanley Cup last season.
Carter wasted little time giving the fans something to cheer for when he scored his 20th goal on a wraparound past Michal Neuvirth only 1:31 into the game.
Neuvirth left at the end of the first period with a lower body injury.
Carter fed Giroux in the slot and he knocked in his own rebound to make it 2-0 with 6:22 left in the second period.
Notes: The Flyers are 2-1 this season vs. the Capitals. ... Knuble played four years with the Flyers and played in 24 playoff games. ... Holmgren had one more year left on his contract.