Crosby, who led the Penguins in the regular season with 62 points (24 goals, 38 assists) in 55 games, gave Pittsburgh a 2-1 lead at 3:47 of the second period with a one-handed redirection of Brian Dumoulin's shot from the point. It was his 190th point (69 goals, 121 assists) in 169 playoff games, tying Brett Hull (103 goals, 87 assists in 202 games) for seventh in NHL history.
Crosby finished with five shots on goal and a plus-1 rating in 25:20 of ice time.
"We had a couple of really good shifts before that, so we had some good zone time," Crosby said. "I was trying to get my stick on it. I didn't know which way it was going to go, but fortunately I was able to put it in a good spot and got my stick on it. It just found its way in there.
"I think it's always good to score in the playoffs, especially early if you can. But I think you try to go out there and do the right things and contribute and do whatever it takes to win. But yeah, it's always great when you can score at this time of year especially."
The goal probably should have been Crosby's second of the game, but at 18:51 of the first period, three seconds after a power play expired, Islanders rookie goalie Ilya Sorokin stretched and got his left skate on Crosby's redirection of a setup by Kris Letang.
Pittsburgh finished 0-for-3 with the man-advantage.
"I think I just hit the side of the net," Crosby said. "But 'Tanger' made a great play there to find me off the side of the net. I thought we did some good things, but yeah, I've got to bury that. If we can continue to put pucks to the net that way and get recoveries like that, we'll generate some scoring chances and the puck will go in the net."
Kasperi Kapanen was able to tie it 3-3 for the Penguins with 3:39 remaining in the third period, 31 seconds after Brock Nelson had given the Islanders the lead, but Kyle Palmieri's second of the game won it for New York at 16:30 of overtime.
Teams that win Game 1 are 490-222 (68.8 percent) winning a best-of-7 NHL playoff series, including 7-1 in the first round last season. Game 2 will be played here Tuesday (7:30 p.m. ET; NBCSN, CBC, SN, TVAS, ATTSN-PT, MSG, MSG+).
"I don't think a lot changed [in the third period]," Crosby said. "You expect a team to push when they're down a goal. I thought that we had some chances to build on our lead when it was 2-1, and we didn't. They were able to hang around. It would have been great to be able to hold on to the lead, but a team's going to push and they did, and we didn't get that big one in overtime."