Anderson turned aside 29 shots over the final two periods as the Senators stymied the Sabres' offensive onslaught and broke a two-game losing streak.
The shutout was the second of the season and 21st career whitewash for Anderson.
Erik Karlsson notched a shorthanded goal and Erik Condra added an insurance tally in the second period for Ottawa.
Ryan Miller saved 29-of-31 shots for the Sabres, who dropped a second straight following victories over Montreal and the New York Islanders.
Notable deaths of 2014 [PHOTOS]