Malkin gave the Penguins a 1-0 lead 1:04 into the game when Jake Guentzel sent a pass near the right post into the slot, where Malkin scored his fourth of the season.
"Our starts have to be better," Islanders coach Barry Trotz said. "We've had to dig ourselves out of a few holes, especially the last three games. Just stay with it; I thought a couple of guys who were having a little bit of tough nights, they stay with it and get rewarded."
Scott Mayfield tied it at 15:12 of the first. Mathew Barzal won a face-off in the offensive zone back to Mayfield, whose shot from the right point deflected off Zach Aston-Reese's stick and past Murray to make it 1-1.
Rust scored 1:09 into the second period with a wrist shot from the left circle on a 2-on-1 to give the Penguins a 2-1 lead.
Beauvillier scored for a fourth straight game to tie it 2-2 at 11:38 of the second. The Islanders were given a four-minute power play after Guentzel was hit with an unsportsmanlike conduct on top of his roughing penalty, and Beauvillier capitalized when Derick Brassard's slap shot from the right circle hit Jack Johnson in front and went to Beauvillier in the left circle, where he quickly shot it past Murray.
"I just lost my cool," Guentzel said. "I've got to be smarter."
It was Beauvillier's sixth goal during his streak.
"I think we have a team that has really cold blood," Beauvillier said. "We stick with it, even though they tied the game late. I think we kind of focus on the process. In overtime, we just want to get it done and get the two points. That's what we did again tonight."