Rust's marker was the Penguins' fourth straight goal. Pittsburgh scored three times in a 7:10 span during the third period.
The Islanders lost for the first time since they fell 5-2 to the Carolina Hurricanes on Oct. 11.
Pittsburgh has snapped the two longest win streaks in Islanders' history. The Penguins also ended New York's 15-game run in 1981-82.
The Islanders dropped to 11-3-1 this season, while Pittsburgh improved to 9-6-1.