Kaliyev made it 2-0 just 1:25 later at 8:05, finishing a 2-on-1 with Rasmus Kupari, and Vilardi gave the Kings a 3-0 lead with a one-timer from the slot at 12:14.
Dobson cut it to 3-1 with a power-play goal at 16:34, scoring on a one-timer from beyond the circles after a feed from Brock Nelson.
The Islanders were outshot 24-13 through the first two periods.
"We made a couple of mistakes, and they made us pay for it," New York coach Lane Lambert said. "I'll give them credit, they played well. They were structurally good and made it very difficult for us on the forecheck."