Will Cuylle scored for the Rangers (8-3-1), who were 0-for-4 on the power play. Artemi Panarin had an assist for his 800th NHL point (274 goals, 526 assists).
"I personally didn't like the effort that we had to attack the game offensively," Rangers coach Peter Laviolette said. "I don't like the fact that we weren't on the attack more."
Igor Shesterkin made seven saves on 12 shots before he was replaced by Jonathan Quick, who made nine saves on 10 shots.
Shesterkin's save percentage plummeted from .933 entering the game to .920 coming out of it.
"It was a bad day," Shesterkin said. "We need to turn this page and be ready for the next game."
Rasmus Dahlin gave Buffalo a 1-0 lead 26 seconds into the first period with a low shot from the left face-off circle that beat Shesterkin through his five-hole.
The play started with Jordan Greenway dumping the puck into the corner in a way so the Sabres could be first to it. They gained possession along the wall and Bowen Byram got it at the right point. He moved it to Dahlin on the left side and he scored from the circle.
"We have definitely been focusing on that just putting it into spots with guys going in with speed," Greenway said. "It helps out a lot when you keep it away from their goalie, who likes to control the puck, and put it in spots where you can get it back. It's helpful. It makes it hard on them. We were able to turn it into something, which got some momentum for us."
Dylan Cozens made it 2-0 at 2:45 of the second period. K'Andre Miller got caught high in the zone and Cozens was open in the left circle. Benson got the puck to him and he ripped a low, far-side shot under Shesterkin's glove.
"First shift we let them walk in and have one, and it didn't get much better after that," Rangers defenseman Adam Fox said. "It was just a sloppy game."
Thompson extended the lead to 3-0 at 11:39, scoring off the rush with a shot from between the circles off a drop pass from Connor Clifton as he drove to the net.
"We talked all night about getting traffic in front of Igor and taking his eyes away, and I thought he couldn't see that one," Thompson said. "There was a lot of bodies in front. We did a good job on that one."