The Rangers were shut out for the first 54:37 of the contest, but rallied to force an extra period and eventually won their fourth in a row.
Grabner scored in the first period, his 10th goal of the season.
Henrik Lundqvist turned aside the other 27 shots he saw to get the win. He has been in goal for all four victories during the Rangers' current streak. Nabokov made 35 saves.