The two clubs also played a split-squad game in Ottawa on Sunday night.
The Islanders doubled their lead on the power play, as Sundstrom deflected Thomas Hickey's point shot past a partially-screened Lehner.
New York pulled ahead 3-0 just 30 seconds later when Johnson fooled Lehner with a shot from 45 feet away that slipped between his right goal pad and blocker.
Puempel put Ottawa on the board in the second period when he pounced on a rebound in front of the net. However, Sundstrom replied shortly after with his second of the night, one-timing home a feed from Joe Diamond.
Tavares and Conacher rounded out the scoring in the third.
Anders Nilsson picked up the victory for the Islanders, stopping 15-of-16 shots in the first half of the game before being replaced by Kevin Poulin, who turned aside all but one of the 16 attempts he faced.
Lehner made 33 saves in defeat.
The Islanders went 1 for 4 on the power play while Ottawa scored once on six chances with the man advantage.
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