After one period of play, shots on goal were 13-12 for the Devils. New Jersey had held an 8-1 advantage in shots at one point before Washington responded.
Pierre-Luc Dubois fired a shot from the right circle past Hellebuyck to put the Capitals in front 2-1 just 2:11 into the second period.
Midway through the second, the Devils tied the game. Brenden Dillon's shot from the left point went off Hameenaho in front and past Thompson to make it 2-2.
"Yeah, it wasn't a pretty one, but I'll take that every day of the week," Hameenaho said. "It was a good shot from Dilly, and I think it hit my glove and went in."
The goal came in Hameenaho's first NHL appearance of the season after he had been recalled from Utica the previous day.
"I think the start of the second period wasn't our best, but we got that one goal, and I felt like after that, we were getting better all the time," he said.
After 40 minutes, shots were 20-19 for the Capitals.
The Devils looked to have gone in front 10 seconds into a power play early in the third as the puck crossed the goal line following a wild scramble in the crease. After an officials' review, however, it was ruled that the puck had been directed into the net with a distinct kicking motion.
A few minutes after the disallowed goal, a shot from the right point by Kolyachonok had eyes, finding its way between the legs of Thompson to put the Devils in front 3-2 at 6:27 of the third. Noesen was in front to provide the screen.
For Kolyachonok, it was a memorable moment in his Devils debut, which also happened to be his third game in as many nights. He had suited up for Utica on both Tuesday and Wednesday before being recalled Thursday.
"I like to play, so I was actually pumped," Kolyachonok said of the demanding three-day stretch.
The defenseman had only arrived in New Jersey earlier in the day but was able to make an immediate impact alongside Dillon.
"That's pro hockey. You got to be ready every time," Kolyachonok said.
Noesen himself contributed on the fourth goal, backhanding the puck in on a breakaway with 2:53 to play.
The veteran forward had a humorous explanation for his finish.
"I just closed my eyes and I was out of breath at the end of that one," Noesen said. "That was probably the fastest I've skated in quite some time. So, yeah, sometimes you get lucky."
Jack Hughes added an empty-net goal with 45 seconds remaining to clinch the outcome.
Hellebuyck stopped 26 of 28 shots to improve to 6-0-0 since joining New Jersey, while the Devils finished with a 29-28 edge in shots.
For Noesen, the contributions from all three recalls demonstrated the depth available to New Jersey.
"We had a couple guys go down for injury, sickness, whatever's going on, and you got guys there that come back, come up and contribute the way we did," Noesen said. "I thought Lenny and I thought Kooley played really well as well. So, yeah, it's nice to see all the boys get rewarded a little bit."
Rodrigues, who finished with a goal and an assist, credited the team's patience after falling behind in the second period.
"We get down 2-1, and we don't force anything, we don't cheat, we don't try to create something out of nothing," Rodrigues said. "We just stick to our game and let our natural talent take over, and that's the way it's got to be."
Keefe acknowledged that it wasn't New Jersey's strongest performance, particularly coming off a demanding stretch of games, but was pleased with the result.
"It's not the prettiest game, but you got to find ways to win these ones," Keefe said. "It's not our best, but two points are hard to come by, so we're definitely going to take them."