Connor McDavid had a goal and two assists to help Edmonton come back from down 3-1 in the third period. He tied the game 3-3 with 1:38 left and goalie Mikko Koskinen pulled for the extra attacker before setting up Draisaitl's winner on a 2-on-1.
"We've done it all year pretty much, coming back from deficits," said Draisaitl, who had an assist. "We believe in here. We just needed to clean up a few things, get a little cleaner and a little more direct towards the net, and I thought we did that in the third and it was a big win for us."
[WATCH: All Capitals vs. Oilers highlights]
Koskinen made 25 saves for the Oilers (8-2-1), who were shut out their previous two games (1-0 shootout loss at Winnipeg Jets on Sunday; 3-0 loss at Minnesota Wild on Tuesday).
"Credit to the group, we battled hard in the third, and it took the full 20 minutes, but we found a way," McDavid said. "They're a veteran team and they know how to win games. They clog it up, they know how to hold leads, and they're comfortable playing in one-goal games, and they showed it tonight. Credit to us for sticking with it and not folding, and battling back."
Video: WSH@EDM: McDavid ties the game late in the 3rd period
Alex Ovechkin scored two goals, and Braden Holtby made 36 saves for the Capitals (7-2-3), whose four-game winning streak ended.
"[Holtby] kept us in the game with saves on a number of odd-man rushes, and to me, those were read-based, and that's something we've got to continue to work on and improve in our decisions," Washington coach Todd Reirden said. "Knowing who you're on the ice against is something we talked about before the game, and I don't have the luxury of having the last change, so you're going to have certain players out there and matchups you don't necessarily like, and players have to understand those reads and choices they make on the ice."
Ovechkin and Washington defenseman John Carlson each extended his point streak to nine games.
Carlson, who had an assist on Ovechkin's first goal to set the record for longest point streak by a Capitals defenseman, leads the NHL with 21 points (five goals, 16 assists).
"Obviously, it was a tough third period for us," Ovechkin said. "We got the lead 3-1, and we just stop playing. We knew they were going to come, and you know with the mistakes we make, we knew we could not give McDavid the speed through the neutral zone, and those mistakes, they will use it and obviously their speed that they have, we cannot give them freedom. Overall, I don't think we played well in first and third."
Video: WSH@EDM: Draisaitl fires shot under Holtby's arm
Darnell Nurse gave Edmonton a 1-0 lead at 13:20 of the first period when his centering pass deflected off defenseman Dmitry Orlov's stick and bounced past Holtby.
Jakub Vrana tied the game 1-1 at 5:40 of the second. He knocked down Nurse's clearing attempt and scored with a turnaround shot from the right face-off circle.
Ovechkin tipped a point shot from defenseman Jonas Siegenthaler at 6:27 to give Washington a 2-1 lead. He has scored in three straight games.
Video: WSH@EDM: Ovechkin deflects Siegenthaler's shot
Ovechkin scored his 250th NHL power-play goal at 13:40 to make it 3-1.
Draisaitl made it 3-2 at 4:25 of the third when he scored off a pass from McDavid.
"We were down 3-1 and had made a couple of mistakes, but we felt like we were playing all right," Oilers coach Dave Tippett said. "We just said, 'Let's stick with it in the third period,' and we got some momentum, we got rolling. We got that first one and we were back in the game. Our big guys kind of took over, and we ended up with the win. After not scoring for a couple of games, the guys came back hard, and we played well against a good team."
Video: WSH@EDM: Ovechkin rips home 250th power play goal
They said it
"If we're going to be the best team, you've got to perform against those players, and we just got outperformed tonight." -- Capitals goalie Braden Holtby on Oilers forwards Connor McDavid and Leon Draisaitl
"It's an important game for us, and it was important coming off that road trip where we didn't get a win. We were hungry to get a win; we don't want to put too many losses together; that gets you in trouble. We had to find a way to get a win, especially playing against a team like that." -- Oilers defenseman Darnell Nurse
Video: EA Sports Overtime Winner: Draisaitl wins it in OT
Need to know
Carlson passed Mike Green, who had an eight-game point streak from Oct. 15-Nov. 1, 2009, and Scott Stevens, who had an eight-game point streak from Feb. 19-March 6, 1988. ... Ovechkin needs one goal to tie Luc Robitaille for 12th in NHL history (668). ... McDavid, who has played 298 NHL games, will become the eighth player in League history to have never gone more than three consecutive games without a point through his first 300 games. He entered the game without a point in three straight. ... Koskinen is 5-0-0 with a 2.35 goals-against average and .927 save percentage.
Capitals: At the Vancouver Canucks on Friday (10 p.m. ET; ESPN+, SN1, NBCSWA+, NHL.TV)
Oilers: Host the Florida Panthers on Sunday (4 p.m. ET; SNW, FS-F, NHL.TV)
Video: Draisaitl, McDavid lead Oilers to overtime win