Quinn cut it to 2-1 at 10:57, taking a flip pass from Peterka and scoring with a wrist shot from the slot.
Peterka tied it 2-2 at 1:41 of the second period, redirecting a feed from Ryan McLeod at the far post.
Quinn gave Buffalo a 3-2 lead with a power-play goal at 8:13.
Michkov tied it 3-3 on a breakaway at 10:36. The 20-year-old forward leads NHL rookies with 24 goals, and has nine points (four goals, five assists) during a five-game point streak.
"He's an exceptional player with the puck, an exceptional player offensively," Shaw said. "I think what he has to learn is that the better he checks and the better he plays without the puck, the sooner he gets it back and then the more opportunistic he can be and the more he can show off the skill package that he has. There's so much room for growth there away from the puck [or] even with the puck. He's going to learn as he goes what works and what doesn't work in so many different scenarios. And he's well down that path already.
"That's a real exciting player to sort of picture where he may be three, four years from now."