Haula cut it to 3-2 at 8:32 on a shot high stick side.
Schenn pushed the lead to 4-2 with a power-play goal at 11:42 of the third period, following up a rebound of Kyrou's shot from the left face-off circle.
"It was a weird little situation there," Kyrou said. "I happened to be pretty open. Good thing I fanned on it. I was trying to go top glove, but it worked out."
The Blues swept the season series with the Devils, winning 5-3 on Jan. 5 in New Jersey.
"They don't give you any easy ice. Real fast team," St. Louis coach Craig Berube said. "You get pressure everywhere … very good transition team, but we did a good job getting back, protecting the middle of the ice in our zone. We handled the rush attack pretty well tonight, I thought."
NOTES:St. Louis is 14-2-1 in its past 17 games against New Jersey. ... It was Binnington's 200th NHL start. He is 111-64-23 in 204 games. … Schenn (three goals, three assists) and Kyrou (two goals, four assists) each has six points in a three-game point streak. ... Thomas has nine points (five goals, four assists) in eight games against the Devils, and five points (one goal, four assists) in a three-game point streak. ... Blues forward Jake Neighbours had one shot and one hit in 12:08 after being recalled from Springfield of the American Hockey League on Thursday. … The Blues placed forward Brandon Saad (upper body) on injured reserve Thursday. ... Devils forward Jack Hughes missed his fourth straight game with an upper-body injury. ... New Jersey forward Yegor Sharangovich had an assist for his 100th NHL point.