"I thought we played really well," Crawford said. "We were keeping them to the outside. We made a couple good plays, we didn't get down after those three goals, had a lot of chances after that. It was good to see our guys come back.
"Overtime was a lot of fun to watch our guys in that one. It was a good win for us. We don't have many games left so we need the points right now, but I thought overall we played a good game."
Zach Bogosian, Vladimir Sobotka, Brandon Montour and Kyle Okposo scored for the Sabres (30-28-9), who trail the Montreal Canadiens by 10 points for the second wild card in the East. Carter Hutton made 41 saves.
Buffalo, which has lost three straight, is 2-7-2 in its past 11 games and has allowed at least four goals in eight of them.
"It's just frustrating," Hutton said. "I can't put it into words right now. It seems like Groundhog Day right now. I do my best to try and compete and it's hard not getting wins here."
Keith gave the Blackhawks a 1-0 lead at 16:01 of the first period on a backhander from the slot.
Anisimov made it 2-0 at 18:26, faking a shot glove side and scoring stick side on a breakaway off a pass from Kane.
Sobotka cut the lead to 2-1 at 3:16 of the second period, tapping in stick side from the left of the crease.