Nielsen, who had three assists in the game, lifted a backhander past Jonan Hedberg in the third round for the deciding score and preserved an elusive win for the Islanders, who led by three goals three times in the game.
Hedberg had 31 saves for the Thrashers. Dwayne Roloson had 22 for the Isles.
New York had a 3-0 lead after one period. The teams traded goals until Atlanta scored three unanswered goals in the second and third periods to send the game into overtime.
The Islanders outshot Atlanta 3-1 in overtime.