Atlanta, which erased two three-goal deficits, lost for the eighth time in a row, but picked up points for a second consecutive night following an overtime loss to Buffalo.
Slava Kozlov scored in the shootout for Atlanta, but Ilya Kovalchuk and Rich Peverley shot wide against Dwayne Roloson to hand New York its third win in five games.
Tambellini scored with a head-on wrist shot against Johan Hedberg, and Nielsen connected on a backhander.
Matt Moulson and Jon Sim scored 2:51 apart in the opening period, and Richard Park, Trent Hunter and Josh Bailey added goals to help New York earn points for the fifth time in six games.
Moulson has scored in all three of New York's games against Atlanta, tallying five goals.
Atlanta twice was down by three goals before leading scorer Kovalchuk blasted a left point shot for a power-play goal 11:23 into the third to tie it at 5. Kovalchuk's 25th goal was Atlanta's third consecutive goal.
It also capped a four-point night for Kovalchuk.
Maxim Afinogenov scored twice in a six-goal second period that saw four Atlanta goals. Nik Antropov and Rich Peverley also scored.
Hedberg made 31 saves for the Thrashers.
Roloson finished with 22 saves.
While Atlanta took advantage of the power play, the Islanders' power play unit finished 0 for 5 to drop to 2 for 45 in the past 11 games.
New York failed to generate a single shot on goal when Blake Comeau drew Christoph Schubert's high-sticking penalty 3:40 into overtime.
The Islanders entered the game with the league's fewest goals scored, but surged to a three-goal lead through the first 18 minutes.