Ovechkin and Richards scored all their goals in a can-you-top-this second period. The teams scored seven times in the second, and Capitals goalie Semyon Varlamov was yanked for Jose Theodore.
Alexander Semin scored twice for the Capitals. Matt Carle had four assists for the Flyers.
Ovechkin and Richards turned the second period into a rapid-fire game of 1-on-1 that gave a sold-out crowd at the Flyers' home opener its money's worth.
After a scoreless first, Richards took a fantastic pass from Carle and shot high from the red line 1:37 into the period for the power-play goal to make it 1-0.
Philadelphia's celebration was short-lived.
The fans were just winding down from the replay when Ovechkin, who scored twice on opening night, scored his first goal thanks to a turnover and tied it only 67 seconds later.
Ovechkin made it 2-2, and Richards had another goal on the power play that tied it at 3-all.
Back-and-forth went two of the top young stars in the Eastern Conference: Ovechkin, the back-to-back Hart Trophy winner, against the Flyers' blossoming young captain.
When Richards scored his third goal of the period only 18 seconds after his second and put it past Varlamov, the crowd burst into cheers. Fans littered the ice with hats and caused a brief stoppage. Richards pumped his fists and was mobbed by teammates after giving Philadelphia a 4-3 lead.
Semin poked his third of the season past Emery, and Kimmo Timonen's first was a wrister through teammate Jeff Carter's screen that put the Flyers ahead 2-1.
No one played a bigger role in Richards' success than Carle. The defenseman forced the turnover that led to Richards' third goal. He set a Flyers record for assists in a period by a defenseman and tied the NHL mark.
Carle had two points in the first two games.
Semin tied it at 4 early in the third, and Brendan Morrison put the Capitals ahead when the puck bounced off him and past Emery as he skidded into the crease.
Scott Hartnell poked in a deflection with 4:10 left in the third that gave the Flyers new life and would sent the game into overtime. The goal was Philadelphia's third on the power play.
Emery, who had a shutout in his Flyers debut, was shaky in his first game in front of the orange-and-black clad fans.
Briere, who missed most of last season due to groin and abdomen injuries, bailed out Emery with his first goal.
NOTES: Washington RW Mike Knuble, who spent the last four seasons with the Flyers, got a nice ovation during pregame introductions. ... Richards' previous hat trick was Feb. 8, 2006, against the Islanders. ... Richards tied the Flyers' record for most goals in a period. ... The last defenseman to record four assists in one period was Los Angeles' Rob Blake, who did it in the second period against Calgary on Jan. 29, 1998. ... Richards was the first Flyers player to post a hat trick in the team's home opener.