Schenn also had an assist, Kevin Hayes had a goal and an assist and Jordan Kyrou and Pavel Buchnevich each had two assists for the Blues (12-9-1), who lost a 3-0 lead before scoring three straight. Jordan Binnington made 42 saves.
"Roller-coaster game," Neighbours said. "That's not how we want to play obviously. We're happy that we got the two points, and we'll move on from it, but there's definitely areas that we can clean up. Just being harder, harder to play against, harder in our battles and stuff like that and just trying to piece together 60 minutes."
Rasmus Dahlin, Peyton Krebs and Zach Benson each had a goal and an assist for the Sabres (10-11-2), who were 3-1-1 their previous five games. Ukko-Pekka Luukkonen allowed six goals on 20 shots after allowing a total of five in his previous three starts.
"We had our chances," Dahlin said. "I mean, we played really good. We could have had five more goals. But they didn't want to go in today. It is what it is sometimes, but I think who got to just look forward. We played really good offense, we created a lot and we can't think too much about this."
The Sabres rallied from down three goals when Dahlin got Buffalo within 3-1 at 13:32 of the first period off a point shot following a Nikita Alexandrov defensive zone turnover.
Benson made it 3-2 at 6:10 of the second period off a face-off win by Dylan Cozens, then a spin and shot through Binnington's pads.
Krebs tied it 3-3 at 10:53 off a long shot by Dahlin from the left point and a big rebound putback.
"I feel like we had them pretty hemmed in there in their zone for long stretches of time," Sabres center Casey Mittelstadt said. "... I think you saw a lot of Sabres hockey out there with the chances and the way we were creating."
St. Louis coach Craig Berube called a timeout after the tying goal and the Blues scored twice in 24 seconds to regain the lead. Hayes made it 4-3 at 13:04 off a pass into the crease by Kyrou, and Schenn made it 5-3 at 13:28 off another pass into the slot by Kyrou.
"He didn't have to say anything special," Schenn said of Berube. "We were going out there and just shooting ourselves in the foot. Turned over pucks and not playing hard enough. Just not tonight, but in general in the second period. That's something we have to change. Can't be giving up 45 shots a night and relying on your goalies so much. Two points, we'll definitely take it. It's a tight league, but we've got lots to clean up here and start tilting the ice a little bit and start taking over games."