Ivan Barbashev scored into an empty net to make it 3-1 with 1:08 remaining, and Schenn's empty-net goal with 25 seconds left made it 4-1.
Johan Larsson scored his first goal with the Coyotes (6-6-1), who had won three straight games, all against the Blues. Darcy Kuemper made 23 saves.
"I thought a couple of guys were a little too cute out there," Arizona coach Rick Tocchet said. "When you win three in a row, sometimes you feel a little too good about yourself. But, for the most part, I'm not too unhappy."
It was the fifth of seven consecutive games between the teams because of schedule changes due to NHL COVID-19 protocols. The first four were played in St. Louis. The Blues won 4-3 on Feb. 2 before losing 4-3 on Feb. 4, 3-1 on Saturday and 4-3 in a shootout Monday.
The Blues and Coyotes are scheduled to play here again Saturday and Monday.
"The compete level is going up every game, and it's kind of becoming like a playoff series," Kyrou said. "Every win is important."
Faulk gave St. Louis a 1-0 lead with a slap shot above the circles on the power play at 2:28 of the first period. He has five goals through 14 games after not having any at the same point last season, his first with the Blues.
"We created some chances, we're competing and working hard," Blues coach Craig Berube said. "These are hard games, they're really hard. … They're really tight."
Larsson tied it 1-1 when he scored off the rebound of a Jordan Oesterle shot from the point at 8:44 of the second period. It was the first goal scored by Arizona's fourth line this season.
Conor Garland scored for the Coyotes with 1:08 left in the second period, but Berube challenged for offside, and the goal was overturned after video review showed Arizona forward Christian Fischer did not legally tag up at the blue line prior to Ilya Lyubushkin entering the zone with the puck with 1:21 remaining.
Fischer, who returned after missing the previous four games with an upper-body injury, was called for interference on his first shift of the game, leading to Faulk's first goal.
"We had a couple of mistakes that cost us the game, things we didn't do in St. Louis," Arizona captain Oliver Ekman-Larsson said. "We've been playing good against them, but we had too many turnovers and mistakes in our own end."
Ekman-Larsson was minus-2 with two shots on goal in 22:22 in his first game since Jan. 16. The defenseman missed the previous 10 games with a lower-body injury.
Coyotes defenseman Niklas Hjalmarsson didn't play after the first period because of an upper-body injury. Tocchet didn't have an update.
Blues forward Oskar Sundqvist left with 3:48 remaining in the third period after he went awkwardly into the boards fighting for the puck with Oesterle. Berube said Sundqvist wasn't seriously injured and should be fine.
NOTES: The Blues won their fourth straight game on the road. … Arizona opened an 11-game homestand, the longest in Coyotes/Winnipeg Jets history. The 2010-11 Tampa Bay Lightning and 1975-76 California Golden Seals each played a 12-game homestand, tied for the longest in NHL history. … Arizona was 4-1-0 in its previous five games. … The Coyotes were 0-for-3 on the power play after scoring four power-play goals in the previous two games (4-for-11).
