In addition to Reinhart, Alexander Nylander, Marco Scandella and Eichel all scored goals for the Sabres. Carter Hutton made 37 saves on 43 shots, his second straight workload of 40-plus shots.
The Sabres were outshot 20-7 during a second period in which the two teams combined for six goals. The Canadiens scored three in a row - from Artturi Lehkonen, Brendan Gallagher and Andrew Shaw - to take a 3-1 lead.
The Sabres tied the game with goals from Nylander and Scandella that were scored 1:15 apart, but Paul Byron capitalized on a turnover by Reinhart at the offensive blue line to put the Canadiens back ahead with 1:57 remaining. Gallagher scored his second goal to add to the lead with 11:25 left in the third.
"In the second period we're trying to do too much ourselves, too much one-on-one hockey," Sabres coach Phil Housley said. "… We come and tie it up, we show some jam, we have the puck on our stick and we just can't make these critical mistakes."
Lehkonen put the Canadiens on the board after Jordie Benn kept prevented a clearing attempt by Casey Nelson from leaving the Sabres' zone. Max Domi collected the puck up high, spun as he approached the net and dished a backhand pass down low to Shaw, who passed across the net to set up Lehkonen.
Their next three goals came in similar fashion. Phillip Danault stole the puck down low and sent a diving pass to the front of the net to set up Gallagher's first goal. Shaw scored on an odd-man rush.
"This is the thing we continue to talk to our players about, is our checking detail," Housley said. "We're going to continue to talk about it because it's so important. When we don't have the puck, you can see, we have to have a bigger commitment without it.
"You're playing against good teams that are fighting for their playoff lives, they have a good transitional game and it ends up in our net, so we've got to continue that message."
Eichel brought the Sabres back within one when he cleaned up a loose puck for his career-high 26th goal of the season. The goal came seconds after the expiration of a four-minute power play, making it 5-4 with 6:47 remaining.
Once again, the Canadiens responded. Max Domi buried a wrist shot on the power play 3:17 later and Tomas Tatar capped the scoring with an empty-net goal.
Reinhart admitted to feeling the frustration of another season without playoffs, but echoed Eichel in saying that pride will be on the line over the final eight games.
"It's not an easy situation to be in," Reinhart said. "When you go through a couple years of it and they start adding up, it just adds to the frustration. When we do have a little bit of success at the start of the year, it makes it that much more difficult to lose, really. We've got to find a way to stick together and start playing for one another down the stretch here."