But in the extra frame, New York's Brian Strait was whistled for a controversial holding call on Pittsburgh captain Sidney Crosby, handing the Penguins a power play.
Thirty-three seconds later Kunitz converted a close-range shot from the left circle off a feed from Crosby, and the Pens grabbed the advantage in their Eastern Conference quarterfinal series.
Kunitz also scored in the first period and added an assist while Crosby finished with three helpers in the victory.
Marc-Andre Fleury turned aside 32-of-36 shots for the Pens.
Okposo added an assist while Casey Cizikas and Matt Moulson also found the net for the Islanders.
Evgeni Nabokov made 20 stops in the loss.