New York had been off since earning three of four points last weekend at Philadelphia and Washington.
"I don't think we expected that great of a start," said Brendan Shanahan, who scored the fourth goal. "Ryan Miller is a great goalie, but all great goalies can have a tough night. He had a tough night combined with the fact we really came out jumping."
Shanahan was at his freshest in quite a while, playing for the first time in six games after leg injuries kept him out.
Henrik Lundqvist, in his first since signing a six-year, $41.25 million deal with the Rangers, made 29 saves. He hadn't played since a loss to Anaheim on Feb. 7, sitting out both games before New York's rare long break.
"It feels like forever when you play a lot and then suddenly you don't play for a few days," Lundqvist said.
Thomas Vanek's power-play goal spoiled Lundqvist's shutout bid with 6:58 remaining.
Miller unexpectedly got most of Saturday off, pulled after the Rangers went up 4-0 after just 12:25 of play. It was his first regulation loss since Jan. 19, before an 8-0-1 run.
"It's on me to make up for that (early goal)," Miller said. "I wasn't ready, either. The first two goals, I got way too rattled. I got too caught up in that. I've got to be better from there to stop the bleeding. Obviously, I didn't."
Avery, Ryan Callahan, Dubinsky and Shanahan did the damage for New York, which leaped over Buffalo in the tight Eastern Conference playoff race by winning for the seventh time in 11 games (7-3-1).
The Rangers moved back into a postseason spot after losing ground this week.
"I don't think where we are dictates what kind of team we have," Dubinsky said. "We've been saying that a lot, but it's time to start acting on it. This was a big start for us after a break that generally doesn't happen during the season and it's just something to build on. We're ready to go."
The offensive onslaught began right after the opening faceoff when Avery scored one second beyond the Rangers' record. During the layoff, Rangers coach Tom Renney switched up his lines, most notably returning Dubinsky to the middle of Jaromir Jagr's trio along with Avery.
Dubinsky drove the net with Jochen Hecht on him. Hecht used a tug that was going to be whistled for a penalty, when Dubinsky got off a shot. Avery was in front and put in the rebound.
"That wasn't the plan. The plan was to give it to me," Jagr said with a laugh.
And how long did that last? "Five seconds," he said.
Callahan made it 2-0 at 4:29, following a turnover by defenseman Brian Campbell. Chris Drury sent a pass in front to Martin Straka, who got a shot on Miller. Callahan knocked the rebound out of the air and in for his second in four games.
Miller had himself to blame for the next two, 3:20 apart.
During 4-on-4 play, Dubinsky raced up from behind his net, traded lateral passes with Jaromir Jagr in the neutral zone, and steamed into Buffalo's zone. Campbell was draped over him, but Dubinsky got off a backhander that beat Miller between the pads and had him looking toward the ceiling.
Shanahan let go a shot from above the right circle that soared over Miller's shoulder, ending his day after four goals on 11 shots.
Thibault didn't face an official shot in the first, but gave up Avery's second of the game -- eighth of the season -- 7:47 into the second.
"We got outbattled in the paint. We got outbattled behind the net," Sabres coach Lindy Ruff said. "It hasn't happened in a long time. You're going to lose some battles but we lost them consistently for the first 10 minutes."
The Rangers' record for fastest goal is 9 seconds, set by Ron Duguay on April 6, 1980, at Philadelphia, and matched by Jim Wiemer on March 27, 1985, at Buffalo. This was their quickest at Madison Square Garden, topping Phil Esposito's goal 12 seconds in on March 1, 1978. ... Buffalo allowed only 13 goals to the Rangers in its six-game playoff victory last season. ... Straka's assist gave him 700 NHL points.