Ray Sheppard's 49th goal on a power play at 10:27 cut Washington's lead to 2-1. Then Keith Primeau beat Beaupre at 18:28 to tie the score.
"The penalty [to Dave Poulin] was tough because we should have iced the puck on the faceoff and didn't," Capitals coach Jim Schoenfeld said.
"We were in good control, and the big guy [Sheppard] got loose and made a big shot. That gave them an emotional boost.
"But if you told me this [Friday] morning I'd come in here and get a point I'd have taken it."
Joe Louis Arena can be a tough place to play, especially with the noise from a standing-room crowd of 19,875.
Beaupre (22-14-7) had played well, making several brilliant stops among 20 saves.
The Caps, 3-1-1 in their past five games, scored on the first two shots they took against Red Wings rookie goaltender Chris Osgood in the first period.
Sylvain Cote ripped a 70-foot slap shot over Osgood's right shoulder at 1:39 to give Washington some momentum. The shot was taken 10 feet outside the blue line.
At 4:55, Kelly Miller gave the Capitals a 2-0 lead when he beat Osgood low on the glove side after taking a pass from Calle Johansson.
Beaupre looked as though he would breeze to his third shutout of the season for most of the first 50 minutes.
"There weren't a lot of shots on goal, but both goalies made some timely saves," Schoenfeld said. "Our whole team played a great game. Detroit is battling for first place [with Toronto], and we're trying to make the playoffs. It was a big game for them and a big game for us."
Primeau knocked Michal Pivonka cold with a booming check early in the second period. Pivonka suffered a gash on the head and didn't return.
Several brawls broke out in the third period and in overtime. The last came with 4.2 seconds to play. Six players, three on each team, received unsportsmanlike conduct penalties.
The Capitals beat the Red Wings 6-3 on Jan. 30 at USAir Arena, Schoenfeld is 14-8-5 since taking over as Washington coach on Jan. 27.
Washington .. 2 ... 0 ... 0 ... 0 ... -- ... 2
Detroit ..... 0 ... 0 ... 2 ... 0 ... -- ... 2