Weber also had an assist after having only four in his first 39 NHL games.
"That's probably the biggest game I have had in hockey, ever," Weber said. "It was pretty special. I guess every squirrel finds a nut sometimes."
Weber helped Miller earn his 200th win in the NHL.
"It's pretty cool to be around long enough to get 200 wins," the 30-year old goalie said. "You never expect to be around that long. I am just happy to be playing and making a career out of it. It's exciting to reach milestones now."
"We had a good third and were strong in the first 10 minutes, but the white part of the Oreo cookie wasn't very good. It was awful," Oilers coach Tom Renney said. "This is the way the season is. As much as we don't like it, we are still going to see this occasionally. What we have to do is limit the number of times this happens moving forward."
Oiler rookie Taylor Hall agreed that the performance wasn't good enough.
"That's certainly not the effort we wanted," he said. "After playing well in the three games on the road we had an opportunity to start the homestand off right and we didn't do that."
Buffalo made it a 3-1 three minutes into the third as Hecht picked the bottom corner on Oilers goalie Nikolai Khabibulin for his fourth goal this season.
The Oilers got back into the game three minutes later when Tom Gilbert threw a shot on net that deflected off the skate of Jones and past Miller.
Miller then made a tremendous save to deny Ales Hemsky during an Oilers power play with eight minutes left in the game.
Weber scored into an empty net with 52 seconds remaining.
Edmonton got off to a quick start in the first as Hemsky made an immediate impact in his return from injury. The skillful winger fought off Tyler Ennis to keep the puck in Buffalo territory and allow Penner to score at 1:27 on the first shot of the game.
Buffalo came close to tying the game midway through the period on Mike Grier's wraparound attempt, but Khabibulin showed impressive speed and came across for a pad save.
The Sabres tied it with just over two minutes left in the first as Patrick Kaleta's wraparound found its way in front to Adam, who caught Khabibulin going the wrong way.
The Sabres went ahead 2-1 early in the second when Weber scored his first with a shot from the point that Khabibulin wasn't expecting.
- The Oilers lost D Ryan Whitney to a right ankle injury early in the first period. Whitney leads the team with 27 points.
- It was the only meeting between the teams this season. Edmonton was 0-3-1 against the Sabres in the past six years.
- Hemsky missed 10 games with a groin injury.
- Edmonton captain Shawn Horcoff skated before the team's practice Monday even though he is projected to be out until February with a knee injury.
- Buffalo was without leading scorer Derek Roy, who's out 4-to-6 months with a quadriceps injury.
- The Sabres were also without C Tim Connolly, out indefinitely after fracturing his nose on Dec. 18. Also missing for Buffalo was D Tyler Myers who missed his first NHL game on Monday in Calgary with an illness.
- Edmonton D Jeff Petry made his NHL debut, giving the Oilers six rookies in the lineup. Petry had an assist.