Nick Spaling also added a goal and an assist, Paul Gaustad potted an empty- netter and Colin Wilson finished with a pair of helpers for the Predators, who have won two straight on the heels of a four-game skid. Carter Hutton made 24 saves in the win.
"It's a good start to the road trip so far," Weber said. "We've got a challenge in Chicago coming up. They're a very good team and we've got to make sure we keep pushing forward and trying to get on a roll here."
Drew Stafford scored the lone goal for the Sabres, losers in four of their last five. Jhonas Enroth allowed three goals on 23 shots in the setback.
"We're not going to have anyone come in here and save us from this. We've got to pick ourselves up. We have what we have and we've got to work through it and we've got to get better," said Sabres coach Ted Nolan.
Stafford opened up the scoring with his power-play marker just 2:09 into the game. Cody Hodgson fired a shot from the left side that bounced off Hutton's pad, but Stafford was there to stuff home the rebound.
The Predators then scored four unanswered goals and Hutton held strong until the final horn.
Spaling's power-play goal knotted things up with less than three minutes left in the first. Weber fired a shot from the right wing into traffic and Spaling was there to redirect it past Enroth to make it 1-1.
Weber scored the eventual game-winner 3:33 into the middle session. With a 3- on-1 rush, Wilson dropped a pass back to Weber, who beat Enroth high for a 2-1 lead.
Smith scored just over four minutes later for a two-goal advantage. Wilson controlled a loose puck behind the net and passed it to Smith on the point, who hammered home a slapshot, his 20th goal of the season.
Gaustad's empty-netter with 1:31 left accounted for the final difference.
2014: The Year in Music [PHOTOS]