The 41-year-old Anderson spent three seasons with the Panthers (2006-09) and still maintains a home a few miles north of their arena. He helped Buffalo get to 64 points and pull into a three-way tie for the final playoff spot in the Eastern Conference with Detroit and Florida.
The Sabres have not been to the Stanley Cup playoffs since 2011 - the longest drought in the NHL.
''The mind just shut off and you worry about the next shot,'' said Anderson, who made 26 saves in the third period.
''Whatever the shot clock got to, it got to. That is about as good a game as I have played in a while. I was good mentally as in being sharp and in the moment. And that is tough to do. This was a night where I did.''
Buffalo also got a power-play goal from Alex Tuch, and Tage Thompson had three assists.
Carter Verhaeghe scored his 31st of the season for the Panthers - who were playing without their two top centers in Aleksander Barkov and Sam Bennett due to injuries. Sergei Bobrovsky made 32 saves in the loss.
Buffalo, coming off an overtime win at Tampa Bay on Thursday night, took the lead at 3:51 of the second period when Skinner found himself alone at the right side of the net and scored on a pass from Thompson.
The Sabres made it 2-0 on Tuch's goal, which came two minutes after Buffalo had a goal taken off the board due to video review showing Tuch was offside.
Florida finally got one past Anderson at 8:24 of the third period as Verhaeghe took a cross-ice pass from Anthony Duclair and fired a shot from the right circle.
''This was a huge home game and it is really disappointing to have the first two periods like that,'' Verhaeghe said. ''We're in a playoff push and there is nothing we can do about it now. We have to be ready moving forward.''
Skinner got his second of the night and 24th of the season about two minutes later to give his team some breathing room again with a tough-angle shot from the goal line to beat Bobrovsky.
''I was pretty upset with them a couple games ago in between periods and said the only thing that matters now is grit and compete and pride,'' Buffalo coach Don Granato said. ''They take that to heart. This is a team with a lot of love for each other and what they're doing and where we're at in our community in Buffalo and they'll fight for it. It's nice to see.''
NOTES: For the first time this season, the Panthers had Duclair in their lineup. He had been out since having Achilles tendon surgery in July. . With Barkov and Bennett out, Florida recalled Grigori Denisenko and Zac Dalpe from its AHL team in Charlotte. Denisenko had been sent to Charlotte on Wednesday to help clear cap space for Duclair. . Florida goalie Spencer Knight will be away from the team for an undetermined amount of time after entering the NHL/NHLPA player assistance program. . The Panthers also placed Givani Smith on long-term injured reserve after he was struck in the face by a puck Monday against Anaheim. . The Sabres made Rasmus Asplund, Jacob Bryson and Eric Comrie healthy scratches.
Sabres: Host the Washington Capitals on Sunday afternoon.
Panthers: Visit the Tampa Bay Lightning on Tuesday night.
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