Fehervary cut it to 4-1 at 5:47 of the third period, scoring into an open net following a pass across the slot by Alexander Alexeyev.
Backstrom made it 4-2 at 17:14 with Kuemper pulled for an extra attacker, but Schenn scored an empty-net goal at 18:21 for the 5-2 final.
"It's obviously not the way we want to play," Backstrom said. "Obviously sometimes you get a nice comeback like last game [5-4 in a shootout against the Buffalo Sabres], but it's too hard. It's too hard in this league to just chase every game. I mean, it's not doable."
Hofer, 22, was recalled from Springfield of the American Hockey League after Jordan Binnington was suspended two games for roughing and unsportsmanlike conduct Wednesday. It was Hofer's third NHL start and first since Nov. 13, 2021.
"There's obviously a lot of uncertainties with being in the AHL all year and the first game in a long time," Hofer said, "but I thought the guys played really well in front of me and kind of kept them to the outside. It's super nice to get the win."
NOTES: St. Louis is 6-0-1 in its past seven against Washington. … Blues forward Pavel Buchnevich (assist) extended his point streak to 11 games (19 points; seven goals, 12 assists), which is the longest by a St. Louis player this season. … Kyrou has a six-game point streak (five goals, four assists). … Capitals forward Dylan Strome saw his NHL career-high six-game point streak end (10 points; three goals, seven assists).