Kyrou made it 2-0 at 6:33 of the second period after stripping New York defenseman Ryan Lindgren of the puck in the neutral zone and scoring on a breakaway.
"I'm just trying to hit his stick a little bit, whatever I can to make him do what he did there," Kyrou said. "It's one on one there and he's skating back for the puck. I don't know if he sees me coming at him with my speed. I just tried to throw him off balance a little bit."
Thomas made it 3-0 at 10:28, beating Quick short side after Blues defenseman Philip Broberg joined the rush for a 2-on-1.
Berard made it 3-1 at 10:18 of the third period after his shot glanced off Blues defenseman Colton Parayko's stick and in under Hofer's blocker.
Cuylle then scored on a shot from the slot at 11:37 to cut it to 3-2.
"It's just hard work," Cuylle said. "Just keeping it simple. A couple rims behind the net, good pinch by the 'D.' A nice play to the middle there and I was just trying to get it on net.
"We've just got to keep fighting, keep battling. I thought we showed a good push there in the last 10 minutes, but that's not enough to win hockey games. You've got to play a full 60 minutes. We weren't able to do that tonight."