Mike Fisher and Ryan Johansen scored goals 4:34 apart in the second period, and Filip Forsberg and Ryan Hartman sealed the victory by scoring in the final four minutes.
The NHL-leading Predators improved to 14-0-1 in their past 15, and also extended their team-best road winning streak to nine straight.
Rinne won his 11th straight to match his personal best in a stretch in which he's allowed just 18 goals. The shutout was the 51st of his 12-year career and third in nine games.
And Rinne also improved to 40-9-4 in becoming the NHL's seventh goalie to win 40 games in a season three or more times.
The Predators haven't lost in regulation since a 3-1 defeat to Detroit at home on Feb. 17. And they improved to 12-0-3 in their past 15 road games since a 3-0 loss at Vegas on Jan. 2.
The Sabres were unable to build off a 5-3 win over Chicago on Saturday and lost in rookie coach Phil Housley's first meeting against his former team. The Hockey Hall of Fame defenseman spent the previous four seasons as a Predators assistant coach.
Buffalo ranks last in the Eastern Conference and had its NHL-worst home record drop to 11-21-5.
Minor-league call-up Linus Ullmark stopped 28 shots, starting in place of Robin Lehner, who was sidelined with an undisclosed injury.
Ullmark had little chance on Fisher's goal that opened the scoring 12:29 into the second period.
Fisher and teammate Scott Hartnell were both parked in front when Fisher deflected in Mikka Salomaki's shot from the blue line. The Sabres challenged the goal by arguing Fisher interfered with Ullmark. The goal was allowed to stand despite a replay showing Fisher nudged the back of Ullmark's skate in the crease and also got a hand on the goalie's glove.
There was no disputing Johansen's goal. Ullmark stopped Viktor Arvidsson's initial shot from the right circle. The rebound fell at Ullmark's his feet where Forsberg chipped the puck to Johansen, who snapped it in the open left side.
The first period was lively with both teams putting their speed on display.
Arvidsson nearly scored 7 seconds in on a breakaway off the opening faceoff.
Buffalo's Jack Eichel burst up the left side and behind the Predators defense to drive across the crease only to have his shot stopped by Rinne's left pad. Rinne also got an assist from Kevin Fiala who got his stick out to stop Justin Bailey's attempt to tap in a bouncing puck in the crease.
