The Islanders evened the score at 5:33 when Anders Lee converted from the right doorstep.
Kyle Palmieri gave the Islanders a 2-1 lead at 18:13 of the period with a breakaway goal.
New York led by one after 20 minutes and took a 17-12 advantage in shots on goal into the first intermission.
Brock Nelson doubled the Islanders’ lead at 8:25 of the second period with a shot from the doorstep.
The Islanders extended their lead to three at 11:14 of the middle frame when Nelson netted his second of the night with a short-handed wrist shot from the slot.
Mittelstadt scored his third goal of the season to cut the Avalanche’s deficit to two with a one-timer from the bottom of right circle off a cross-ice pass from Nathan MacKinnon at 37 seconds of the third period.