Sam Carrick, Adam Henrique and Troy Terry scored for the Ducks (10-18-0), who have lost four straight and 12 of their past 13. Gibson made 30 saves.
“We sleepwalked through the first period. Kind of saw that coming, and we kind of got traction as the game went on," Anaheim coach Greg Cronin said. "I thought we played a solid third period, and we just fumbled the power play at the end. There’s not much to say.”
Palmieri gave the Islanders a 1-0 lead 40 seconds into the second period, beating Gibson short side with a one-timer from the right circle off a pass from Brock Nelson from behind the net.
Cizikas made it 2-0 at 6:32 from the left circle off a return cross-ice pass from Cal Clutterbuck on a 3-on-2 rush.
Terry cut it to 2-1 at 7:47 when his cross-ice pass redirected into the net off the skate of Robert Bortuzzo on the power play. It was his first goal since Nov. 1 (18 games).
Henrique tied it 2-2 at 13:29. Urho Vaakanainen broke his stick on a one-timer from the left circle, but the puck deflected off the skate of Jean-Gabriel Pageau to the edge of the crease, where Henrique tapped it into an open net.
Carrick gave the Ducks a 3-2 lead at 5:07 of the third period with a shot off the rush that hit the stick of Samuel Bolduc and knuckled past the blocker of Varlamov.
“I was trying to shoot through him and use him as a screen,” Carrick said. “I think it hit off him and kind of got a lucky bounce. Anytime you can get a chance in the slot there to throw it on the net, it’s never a bad shot.”