Casey Mittelstadt had a goal and an assist, and Devon Levi made 29 saves for the Sabres (42-33-7), who won five of their final six games.
"It's been a long year," Skinner said. "Obviously we didn't reach the goal we wanted to, but you have to be professional and put a good effort in. We had a lot of chances early. Their goalie made some good saves. We had to stick with it, and it paid off."
Tyler Angle scored his first NHL goal in his second NHL game for the Blue Jackets (25-48-9), who were 1-4-1 in their final six games and finished 31st in the NHL.
"It's the end of the season and we know we're out. It's a difficult game all the way around," Columbus coach Brad Larsen said. "This is a privilege to do this job. It is. And I love it, even in a challenging year like it was."
Jon Gillies allowed three goals on 27 shots in the first two periods before leaving with a lower-body injury. Michael Hutchinson made seven saves in relief.
Gillies was another example of the injury-ravaged season for the Blue Jackets, who played most of it without defensemen
Zach Werenski
(shoulder) and Jake Bean (shoulder), and forward Justin Danforth (torn labrum).
Columbus finished with a franchise-record 563 man-games lost (508 in 2014-15).
"This has been the hardest season with all the injuries," Blue Jackets forward Johnny Gaudreau said. "It's hard to see all the guys out for the season and out six, seven weeks. It's hard to watch teammates like that go down, especially when they're a huge part of the team."