Ottawa has captured seven of its last eight and stayed within three points of Boston for the lead in the Northeast Division.
Anderson made 34 saves in the 17th shutout of his nine-year career. He helped halt the Rangers' five-game winning streak.
Jason Spezza scored twice, the first goal coming 8:54 into the second period that broke a scoreless tie.
Milan Michalek tallied with 6:02 to play in the third and Spezza added a power-play goal less than 2 minutes later.
Henrik Lundqvist allowed the three goals on 24 shots.