The shot sneaked in under Enroth's left arm and had enough to dribble in behind him.
Enroth was nearly the difference in stopping 37 shots, including Kane's first-period penalty shot.
Enroth bounced back after both he and Ryan Miller allowed four goals apiece to the Rangers.
Enroth was in position to stop Wayne Simmonds' shot from the slot with 50 seconds left.
For Enroth, it was his third career shutout and first since, Nov. 18, 2011.
Sabres coach Ron Rolston disclosed after the game that Enroth was cramping the entire third period.
Mike Ribeiro nearly sealed it for Washington 2:46 into overtime, but was robbed by Enroth's diving save.
But Richards made it 8-3 when his shot from roughly 40 feet eluded Enroth, completing his hat trick.
Alfredsson then took a snapshot from the slot off a feed from Mika Zibanejad that beat Enroth cleanly.
Phillips took a point shot that dribbled across the goal line after Enroth got a piece of it.
Enroth kicked out a long rebound and Elias had a wide-open net with 66 seconds to play in regulation.
Enroth, who stopped 35 shots, then gave up goals on both shootout attempts.
Enroth was especially sharp in the third period, when he made four of his saves during the Rangers' only power play.
Enroth lay prone on the ice after the play and had to be helped to the bench, dragging his right leg.
Karlsson opened the scoring at 8:17, just 27 seconds into the game's first power play when his point shot eluded Enroth.
Enroth got just enough of his glove on Kane's shot to have the puck deflect off the crossbar and over the net.
Enroth was briefly shaken up on the play, after teammate Mark Pysyk rolled over top the goalie while chasing Gagne from behind.
Armstrong, with his second goal of the season and second in two games, deked to the backhand to beat Enroth at 16:09.
The shot stunned Weber and Henrique was able to get the loose puck and fire it past Enroth from the left circle.
"I felt very confident and in control of every shot, " Enroth said.
Enroth stopped 11 shots in the final period, including getting a piece of Wayne Simmonds' shot from the slot with 50 seconds remaining.
After Holtby got his right pad out to stop Buffalo's Tyler Ennis, Ovechkin sealed the victory by snapping a shot through Enroth's legs.
They fell behind 2-0 midway through the third period, and the way Sabres starting goaltender Jhonas Enroth was playing, things didn't look promising.
The Canadiens had an 11-2 shot advantage in the second period but failed to score, with Jonas Enroth stopping Michael Ryder at the doorstep.
Jhonas Enroth made the start in place of Ryan Miller and is still searching for his first win in more than a calendar year.
Much of the Flyers offensive struggles on Saturday had to do with Enroth, who earned his third career shutout and first since Nov. 18, 2011.
Enroth was outstanding in making his first start since Feb. 5.
Enroth was looking for his first win since November 2011 (0-10-3).
Just 32 seconds after Gerbe's goal, Callahan stole the puck from Buffalo's Mike Weber and his forehand beat Enroth on the glove side for a 7-2 lead.
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