The Islanders, fighting to climb out of the cellar in the Metropolitan Division, won its third in a row.
Semyon Varlamov had a shutout working with time running out, but Nelson's wrist shot got the best of the Colorado goaltender with 4:28 to go in regulation.
Grabner's goal came 1:46 into overtime on what was New York's 24th shot of the game.
Nathan MacKinnon scored Colorado's goal with 1:45 to go in the middle period.
The win went to Kevin Poulin, who stopped 30 shots.