Gostisbehere, uncovered in the slot, put the Flyers ahead 3-2 at 3:47 of the third period.
"We had a nice 3-2 lead. Blew a 2-0 lead, then had a 3-2 lead and we just let it slip away," Gostisbehere said. "It definitely hurts right now. I don't know what else you could do. We've got to move on and get some points in these next games."
Skinner's tying goal went in off his skate and was upheld after a video review. Asplund broke the tie when he put in a rebound 25 seconds later.
"We've had a much better belief system progressively the last couple weeks and it's getting stronger," Sabres coach Don Granato said. "You can feel that down the bench, up and down the bench. More confident speak, less emotional speak. So, yeah, a lot of subtlety but good signs."
Vigneault said, "We made some mistakes that we just gave them goals. Three of them. That face-off goal and obviously the third and fourth goal. Things that you can't do in this League and things that we did tonight with some of our most experienced players."
NOTES:The Sabres scored three or more goals in the final five minutes of regulation for the 11th time in their history. … The 25-second span is the Sabres' fastest two goals in a game this season; their previous best was 37 seconds against the Flyers on March 31. … It's the sixth time in Buffalo history that the Sabres have won in regulation when trailing in the final four minutes of a game. … Hart, who played in his 100th NHL game, is 0-5-2 in his past seven games. ... The Flyers (5-3-0) won five of the eight games against the Sabres (3-3-2) in the season series.