Copley entered the game and kept Edmonton from adding to the lead while Washington rallied.
"I thought the team played really well in front of me defensively," Copley said. "There were some rebounds I gave up that guys were clearing."
Eller took a pass from Justin Schultz at the point, deked Koskinen and slid a backhand in at 7:36 to pull the Capitals within 3-1.
Sheary made it 3-2 when he scored off a rebound at 9:16 of the second period.
The Capitals tied it 3-3 at 2:48 of the third period when Kuznetsov's shot deflected off Oilers defenseman Tyson Barrie.
"Just got a little unlucky at the end and I felt like they felt a little more fresh, and you can feel that," Kuznetsov said. "We had a tough game last night, and that is not an excuse, but at same time I like the way we responded after 3-0."
NOTES:Kane has scored 28 points (21 goals, seven assists) in 35 games against the Capitals. He has scored two goals in three games since signing a one-year contract with the Oilers on Jan. 27. … Oilers defenseman Tyson Barrie played 13:52 and took a penalty in his return after missing five games with an upper-body injury. … Draisaitl enters the All-Star break one goal behind Chris Kreider of the New York Rangers for the NHL lead. … Capitals forward Tom Wilson will replace Ovechkin on the Metropolitan Division roster and will make his NHL All-Star Game debut. … Capitals forward Nicklas Backstrom had an assist and has scored 13 points (three goals, 10 assists) in 14 games this season. … Nugent-Hopkins is the third Oilers player to score a go-ahead shorthanded goal in the final five minutes of regulation, joining Magnus Paajarvi (16:50, Feb. 10, 2013) and Paul Coffey (16:46, Dec. 26, 1984). … Edmonton scored on its only power play after it was 0-for-10 the past four games. … Copley was reassigned from Hershey of the American Hockey League to the taxi squad on Wednesday.