The Sabres offense went from managing just three shots in the first period to a six-goal third period, with Tage Thompson scoring a pair on the power play. Victor Olofsson and JJ Peterka scored goals 13 seconds apart, and Alex Tuch and Peyton Krebs also scored.
Dylan Cozens had three assists and the Sabres improved to 6-3-2 in their past 11 since snapping an eight-game skid. Thompson's two goals pushed his total to 23 on the season. He entered the day third in the NHL scoring race.
The shutout was the 41-year-old Anderson's 43rd of his career, and first since playing for Ottawa, when he stopped 35 shots in a 4-0 win over Buffalo on March 26, 2019.
In crediting the entire team for the shutout, Anderson was impressed with how Buffalo didn't show signs of frustration following the first period in which the Kings were intent on bottling up the middle of the ice.
''We could come in here after having three shots on goal and be frustrated, but the message wasn't being frustrated, it was `Get back to our game, do what works,''' Anderson said.
''I think a lot of times younger teams that haven't played the right way for too long, if they get off their game, they start trying to do too much, be individual,'' he added. ''We didn't do that tonight, which is a huge steppingstone in our growth.''
Pheonix Copley stopped 25 shots for the Kings, who dropped to 5-6-4 in their past 15. They also allowed four or more goals for the sixth time in their past 11, including a 6-5 overtime loss at Columbus on Sunday.
Copley allowed six goals on 16 shots in the final frame in making his third start since being called up from the minors following Cal Petersen's demotion on Dec. 1.
''Responsibility. Carelessness,'' Kings coach Todd McLellan said in summing up the loss. ''Been talking about it all year. No, I can't explain it.''
What bothered him most is how the Kings got away from their defensive structure by taking too many penalties, which led to Buffalo scoring three power-play goals.
Thompson opened the scoring 1:21 into the third and two seconds after Carl Grundstrom was penalized for slashing to give Buffalo a brief two-man advantage. Cozens beat Kings captain Anze Kopitar on the faceoff to the left of the net and drew the puck back to Thompson, who snapped in a shot inside the left post from the high slot.
The rout was on when Olofsson one-timed in Rasmus Asplund's centering pass to make it 2-0 at the 9:56 mark, followed by Peterka completing a give-and-go on a 2-on-1 rush with Jack Quinn.
''I think we did a really good job. I thought we didn't force really anything,'' said Thompson, who a week ago scored five times in a 9-4 win at Columbus. ''I thought we were patient, waited for the right time and waited for the right shot and it paid off.''
The Kings' best scoring chance came late in the first period, when Gabriel Vilardi was set up alone to the left of the net but fanned on a shot that dribbled behind Anderson and rolled through the crease.
Kings: Close six-game road swing at the Boston Bruins on Thursday.
Sabres: Open three-game road trip at the Colorado Avalanche on Thursday.
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