Kyrou tied it 1-1 at 4:02 with a snap shot from the top of the right circle off a cross-ice pass from Thomas.
He then made it 2-1 with a power-play goal at 7:30, putting the puck in at the right post after a tic-tac-toe passing play with Thomas and Buchnevich.
"'Tommer' hit me with speed coming wide and I dropped it to him, and he made a great play. 'Buchy' found me for a tap in there," Kyrou said. "I think that was a big goal for us, especially a power play goal, big power play goal to get the lead there."
Brandon Saad's power-play goal at 9:26 of the second period made it 3-1, a one-timer from the right circle.
Kyrou completed the hat trick to make it 4-1 at 7:01 of the third period on a breakaway.
Trocheck cut it to 4-2 at 10:58 on the power play with his fourth goal in the past three games. He beat Binnington on a one-timer from the bottom of the right circle off a pass off the end boards from Artemi Panarin.
"We had tons of chances," Trocheck said.
Buchnevich scored an empty-net goal at 19:42 for the 5-2 final.
"It was a clean game, fun," Binnington said. "You know you're going against a good offensive team over there. It was a fun working night."
NOTES: It was Kyrou's third NHL hat trick. He has eight points (four goals, four assists) in five games against the Rangers. ... Panarin has 24 points (eight goals, 16 assists) in 21 games against St. Louis, and 15 points (nine goals, six assists) during a nine-game point streak; his three-game goal streak ended. … Trocheck has eight points (four goals, four assists) during a four-game point streak.