"He amazes me every night I come to the rink," DeBoer said of the 41-year-old forward. "I don't have a lot more adjectives to describe him, but he's a pleasure to work with."
It was Jagr's 13th goal of the season and career No. 694, moving him into a tie with Mark Messier for seventh all-time.
Greene, who knocked the rebound of a shot by Jagr past Braden Holtby for the game-winner, also had two assists.
"I honestly can't remember the last overtime goal I scored," Greene said.
Dainius Zubrus had a goal and an assist for New Jersey, which trailed 3-1 after two periods.
"I think everyone's surprised we came back [but] I think the least surprised guys are in our room," DeBoer said. "I think they felt confident all night they could still make a game of it."
Martin Brodeur made 18 saves for the Devils, who lost in overtime to Anaheim on Friday night.
"Today, I felt like we were playing very good hockey the first two periods, and I look at the scoreboard and we were down 3-1," Jagr said. "That was kind of tough to swallow, but we got some lucky bounces in the third period, and tied the game and went into overtime again.
"I thought we responded very well when we were down two goals, and it's a big win for us," he said.
Holtby finished with 32 saves in the first meeting of the season between Metropolitan Division rivals.
"They are a relentless team. They kept coming," Capitals coach Adam Oates said. "We made a couple of mistakes."
Jagr, on a power play, and Ovechkin traded third-period goals three minutes apart. Zidlicky pulled the Devils within 4-3 when he beat Holtby from the slot at 9:21.
New Jersey tied it when Travis Zajac intercepted Holtby's clearing attempt and fed in front to Zubrus, who poked it home at 11:47.
"We just gave them a little momentum to get back into it and they capitalized on their chances right to the end," Ward said.
Both teams were playing the second of back-to-back games and there was little action early on.
Brodeur made several good stops during a Washington power play midway through the period. New Jersey got the man advantage when Tom Wilson went off for high-sticking at 10:52.
Less than a minute later Greene fed Zidlicky in the left circle and Zidlicky one-timed a drive past Holtby on the short side to make it 1-0. It was New Jersey's first shot on goal.
The Devils tested Holtby several times early in the second period, but he kept the deficit at one.
Chimera finally tied it at the 10:07 mark when he came around the back of the net and tried to stuff the puck home. Brodeur made the initial save, but Chimera poked the loose puck past him.
Grabovski gave Washington the lead at 14:38 of the period with his 10th of the year, off a feed from Eric Fehr in the left circle.
"We kind of shot ourselves in the foot there in the second period," Greene said. "There were a couple of breakdowns and they capitalized."
The Capitals went ahead 3-1 with 45 seconds left in the period. They skated in three on two and Mike Green fired from the right side. Brodeur made the save, but it went right to Ward who put it past him for his 11th of the season.
"We played a good 40 minutes," Green said, "but we needed 60 tonight."
Ovechkin has scored 30 or more goals in nine straight seasons. He's scored a goal in four straight games. ... Jagr extended his points streak to seven games with a first-period assist. ... The Devils were without C Andrei Loktionov (illness) and RW Damien Brunner (knee), who was injured Friday against Anaheim. The Devils said Brunner will be out a minimum of four weeks. ... Olympic swimming gold medalist Michael Phelps was in attendance.