Conor Sheary scored, and Darcy Kuemper made 28 saves for the Capitals (34-32-9), who have lost five of six (1-3-2) and are five points behind the Penguins for the second wild card, having played one more game.
"It's tough that we lost a point," Washington coach Peter Laviolette said. "A point's a big thing right now. It's frustrating. It was tight out there. We knew it was going to be that way. We expected this type of game, low scoring, not a lot of shots, not a lot of chances."
New York outshot Washington 15-5 in the third period.
In overtime, Sorokin stopped Alex Ovechkin's backhand in front with 2:12 remaining, and Kuemper made a save on Noah Dobson's wrist shot from the slot with 1:57 left.
"We didn't give up a ton of chances and when we did, [Kuemper] came up big for us," Capitals defenseman Trevor van Riemsdyk said. "I thought we did some good things and had some chances, but both goalies played well. Unfortunately, it came down to a shootout and you never know with those."