Evan Rodrigues scored, and Carter Hutton made 22 saves for the Sabres (23-22-7), who ended a three-game losing streak.
"It's a big win for the group," said Eichel, who scored for a third straight game. "I thought we hunkered down, it's a good hockey team. Obviously, they're fighting for points right now, too. They're arguably one of the hotter teams in the League right now. Give them credit, I thought they played really well. It was good to see us find a way to get a win. It was a huge goal by Roddy in the first and we made some big plays when we needed to, Hutts had some huge saves and we find a way to win this."
WATCH: [All Blue Jackets vs. Sabres highlights]
Hutton got his first win since Oct. 22. He had been 0-8-4.
"It's obviously great to get the win," Hutton said. "Like I've said a few times, there's obviously a lot more going on than what everyone knows with the information and stuff but I just try to stick with it. I've played for a long time here, it wasn't like I just fell off a cliff here."
Oliver Bjorkstrand scored, and Matiss Kivlenieks made 22 saves for the Blue Jackets (27-16-9), who lost for the second time in their past 10 games (8-2-0). It was their first game since a 4-3 win against the Winnipeg Jets on Jan. 22.
"We'll take the point and go. That's a big point for us, coming back in the third period," Blue Jackets coach John Tortorella said. "I thought we fought the puck at times."
Bjorkstrand scored at 5:45 of the third period, picking up a deflected pass in the slot to tie the game at 1-1. It was his team-leading 17th goal of the season, including seven in his past four games.