Hutton had made just two saves when Julien Gauthier toppled onto him in the game's opening minutes and he was forced to leave the game. He was helped off the ice by the training staff, favoring his left leg as he exited through the Zamboni doors to his left.
Tokarski made eight saves in the first as the Sabres were outshot 10-5 in the period.
Buffalo's best chance came with under 2 minutes remaining when Jacob Bryson was stopped on a breakaway and drew a hooking penalty to put Buffalo on the power play with 1:34 left in the first. Bryson's shot was the Sabres' third of the period.
Kinkaid robbed Skinner twice in the first 1:14 of the second. Jacob Trouba hit the post for the Rangers moments after that second save.
Asplund opened the scoring at the 3:52 mark of the second period when he crashed the net and jammed the puck in on the short side. Asplund was called up from the taxi squad before the game.
New York tied things up at the 7:13 mark when Kakko put home his own rebound. He accepted a feed from Artemi Panarin from behind the net after Panarin won a battle off a faceoff in the Sabres zone.
With 9:21 left in the second and 4 seconds remaining on a 5-on-3 power play, Kreider redirected in a pass from Panarin to put the Rangers ahead.
They extended their lead to 3-1 when Fox popped in a rebound with 1:42 remaining. Fox kept driving to the next on the rush and was all alone to knock in the second chance on Mika Zibanejad's blast from the slot.
The Sabres rallied with a pair of goals early in the third period.
Cozens put Buffalo within one when he roofed a tight pass from Sam Reinhart 1:09 into the third period. The Sabres rookie made his return to the lineup after missing the past four games with an upper-body injury.
Skinner made it 3-3 at the 4:22 mark. He accepted a long stretch pass from Brandon Montour and beat Kinkaid high blocker, in just about the same spot Cozens scored the fourth of his career.
Kreider netted his second of the game when he scored on a rebound 5:47 into the third to give the Rangers a 4-3 edge.
Kakko buried an empty netter with 57.7 seconds remaining in regulation.