Nelson made it 2-1 at 5:06, scoring a power-play goal on a one-timer from the crease after Dobson intercepted a Ryan O'Reilly outlet pass to defenseman Niko Mikkola, who came out of the box after the Blues killed a two-man advantage.
Josh Bailey made it 3-1 at 7:05. After a Thomas neutral zone turnover, the Islanders came into the zone with an odd-man rush, and Bailey poked a puck at the right post off St. Louis defenseman Nick Leddy's stick.
Lee made it 4-1 at 10:18 when Nelson, who picked up his 400th NHL point, skated around the net and found the Islanders captain alone in the slot.
Tarasenko gave the Blues a 1-0 lead at 8:17 of the first when he lifted a shot off the stick of Islanders defenseman Alexander Romanov from the right side of the net. Tarasenko collected a puck after defenseman Torey Krug's shot came off the back boards around the other side of the net.
"Loved the first period, especially the first 10 minutes, 12 minutes," St. Louis coach Craig Berube said. "Come out in the second, it's a goal that there's no reason for it to happen."