Palmieri took a pass from Jean-Gabriel Pageau low in the right face-off circle and beat goalie Tristan Jarry over his left shoulder and off the post. Palmieri and forward Travis Zajac, who was a healthy scratch, were acquired in a trade with the New Jersey Devils on April 7.
"It's an incredibly exciting time," Palmieri said. "The opportunity to be in the [Stanley Cup] Playoffs is something I don't take for granted. Obviously, it's hard to get here. We came on the road on a mission. We were able to go out there tonight and find a way to get a win."
Pageau had a goal and two assists for the Islanders, who are the No. 4 seed in the MassMutual East Division. Rookie Ilya Sorokin made 39 saves starting in place of Semyon Varlamov, who did not play for precautionary reasons after sustaining "a little strain" in New York's regular-season finale Monday.
"We had to make a decision, really, last night," Islanders coach Barry Trotz said. "[Varlamov] skated both days. He's going to practice tomorrow as well. We just thought, long term, that's one thing that we think we have is two capable goalies. If we want to have any success, we're going to need both of them. So we said, 'You know what? Let's sort of do what's right. Let's not risk anything here. Let's be cautious. We'll give him another day of practice, and then he'll be 100 percent.'"
Sidney Crosby and Kasperi Kapanen scored, and Jarry made 37 saves for the Penguins, who are the No. 1 seed. Pittsburgh played without forward Evgeni Malkin.
"I thought Tristan made some big saves throughout the course of the game to keep the game where it was at," Pittsburgh coach Mike Sullivan said. "There were some good saves at both ends of the rink. I just think we have to do a better job as a group."
Game 2 of the best-of-7 series will be played here Tuesday (7:30 p.m. ET; NBCSN, CBC, SN, TVAS, ATTSN-PT, MSG, MSG+). 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.
Brock Nelson shot under the glove of Jarry to give the Islanders a 3-2 lead at 15:50 of the third period, but Kapanen tied it 3-3 31 seconds later with a wrist shot off the left post at 16:21.