Despite playing well throughout long stretches of the game, the Wings yielded a 1-0 lead at 19:10 of the second period and a 3-2 lead at 18:59 of the third.
Alex Ovechkin put it away for Washington with a power-play goal at 1:56 of overtime.
His 10th goal of the season came after a possible goal-saving play by Trevor Daley, stopping another chance by Ovechkin, but drawing a tripping penalty.
The Red Wings played well but they gave the game back twice.
Then Tomas Tatar went to work.
At 12:45 of the third period and on the power play, Henrik Zetterberg sent one of his fine passes across the front of the goal mouth to Tatar, low in the faceoff circle to the left of the Capitals goalie Braden Holtby.
Tatar blasted a shot that beat Holtby cleanly after he tried to move quickly across the crease, tracking the pass.
The Wings entered the period down a goal, despite some good play and high energy for much of the game. But Tatar took care of that, too, tying the contest at 2 on a wild goal.
With the puck on Dylan Larkin's stick on the left wing near the blue line, he suddenly spun and fired a hard wrist pass down low in the left circle.
Waiting there, Tatar thrust out his stick and deflected the shot 20 feet, at a 45-degree angle, up over goalie Holtby's left shoulder. Holtby, seeming as surprised as anyone, flailed with his catching glove to no avail.
They were Tatar's second and third goals of the season.
But T.J. Oshie's power-play goal at 18:59 of the third tied it.
The Red Wings played well in the first period, matching and at times exceeding the Capitals' energy and outshooting them for much of the period until Washington nipped them at the end, 14-13.
No goals were scored.
Mrazek stopped a few terrific chances, including a two-shot sequence at the end of Riley Sheahan's high-sticking penalty at 9:51.
In short order, bang-bang, Mrazek extended himself to stop Andre Burakovsky and then Evgeny Kuznetsov.
Then 4:31 into the second period and on a Capitals power play, Darren Helm intercepted an intended cross-ice pass from Ovechkin, on the point, and in the space of a stride or two, Helm flew up ice.
He fired a 20-foot wrist shot from the left wing, after entering the face-off circle that beat the Capitals goalie Holtby cleanly.
It was Helm's first goal of the season and the Wings' second short-handed goal of the season.
Mrazek continued to play marvelously, including a courageous save on Ovechkin, who with skating momentum behind the shot fired a slapper from about 20 feet that hit the goalie square on the chest. He went down, briefly, but popped right back up.
Then the Red Wings allowed a last-minute goal in the period, after playing well.
It was symptomatic of a lot of their failure last season.
After a faceoff in the circle to the right of Mrazek, Burakovsky got the puck back at the point and fired a shot through a player or two screening the goalie and it went in.
Dmitri Orlov and Tom Wilson assisted.
Early in the third period, with the Red Wings on a power play, the Capitals scored short-handed.
A blocked shot wound up on Alex Chaisson's stick and he lifted a high pass out of the zone that Jay Beagle swept up. He broke in on Mrazek on a breakaway and fired it by.