The teams were playing against each other for the first time since the 2020 Stanley Cup Playoffs, when the Islanders won the best-of-7 Eastern Conference First Round in five games.
They will play again here Thursday (7 p.m. ET; NBCSWA, MSG+ 2, NHL.TV).
"This is playoff mentality right now, and I'm not quite sure that we're in our playoff mentality right now, which is really [ticking] me off," Islanders coach Barry Trotz said.
New York took a 2-1 lead at 10:01 of the second period when Washington defenseman Zdeno Chara turned the puck over to Barzal, who scored on a backhand shot from just inside the right circle.
Sprong, acquired in a trade with the Anaheim Ducks on Feb. 24, 2020, scored his first goal with the Capitals from the high slot at 17:21 of the second to tie it 2-2.
"I think our line is creating a lot of chances over the last four games and we knew it was going to come any time now, but [Daniel] Carr made a great play to the middle and I got a shot off," Sprong said.