Clarke took a drop pass from Adrian Kempe in the right circle and roofed a shot over the glove of Semyon Varlamov.
Artemi Panarin scored twice for the Kings (2-4-2), who have won two straight after losing their first six games to start the season. Darcy Kuemper made 41 saves.
Victor Eklund and Matthew Schaefer scored for the Islanders (5-3-2), who extended their point streak to five games (3-0-2). Varlamov made 24 saves.
Panarin gave the Kings a 1-0 lead at 5:51 of the first period. He scored on a rush from the right circle over the glove of Varlamov, who was cheating for a cross-ice pass.
Panarin then scored his second goal of the period following a scramble near the crease to make it 2-0 at 16:37. Panarin's initial shot from the left circle was blocked by Matthew Kessel, but the rebound popped right back to him and he beat Varlamov under his glove arm.
The goal came after Mikey Anderson slid along the goal line to stop a shot from Kessel from crossing the goal line at 15:51.
Kempe had a chance to extend the lead on a penalty shot after Schaefer hooked him on a breakaway at 10:59 of the second period, but Varlamov made the save with his right pad.
Eklund scored on a power play to cut the Islanders' deficit to 2-1 at 14:06. His one-timer from above the left circle trickled under the left pad of Kuemper.
The Islanders outshot the Kings 32-15 through two periods.
Schaefer tied the game at 2-2 at 7:06 of the third period, skating into the right circle and beating Kuemper blocker side.
Kuemper kept the game tied with 52 seconds remaining in regulation when he denied Eklund with his right pad after Schaefer's shot produced a rebound.
