Facing a 1-0 deficit, Clarkson put the Devils on the board with 1:39 to play when his turnaround shot from the low right circle snuck between the left arm of Anders Nilsson.
Moments later, Zidlicky lined up a slap shot from the right point that blew past a screened Nilsson and the Devils kept the Islanders away the rest of the game.
Johan Hedberg turned aside 23 shots for the Devils, who have won three straight.
New Jersey holds the sixth spot in the Eastern Conference with 83 points.
John Tavares scored the lone goal of the game for the Islanders, who finished a string of three straight games against the Devils with one win and two losses. Nilsson made 33 saves.
Nilsson had stopped 21 shots through the first two periods and Hedberg had turned aside 16, but the Islanders got on the board 6:46 into the third.
Matt Moulson had the puck along the deep left boards and pass it behind the net to P.A. Parenteau, who sent a one-touch pass to the front where Tavares one-timed it home for his 27th of the year.
The Devils host Philadelphia on Sunday...The Islanders play in New York against the Rangers on Sunday...New Jersey has won three of five against the Islanders this season.