Jack McBain scored into an empty net with four seconds remaining for the 4-2 final.
Michael Kesselring and Matias Maccelli also scored for Utah (6-5-3), which had lost six of seven (1-4-2) and five straight on the road (0-3-2). Connor Ingram made 13 saves.
"We had a few good games in the past few, but we didn't get the points," Utah coach Andre Tourigny said. "It was important tonight to find a way to get the two points, especially when they tied it in the third. I think we bounced back."
Kyrou scored for the third straight game, Parayko also scored, and Pavel Buchnevich had two assists for the Blues (7-7-0), who had won two in a row. Joel Hofer made 27 saves.
"We were just a little sloppy with the puck," St. Louis forward Brandon Saad said. "They do a good job of clogging up the neutral zone, they've got a lot of speed and skill going the other way. I think we just got stubborn with it, didn't get pucks in deep."
Kesselring gave Utah a 1-0 lead 18 seconds into the first period when his breakaway attempt slid through Hofer's legs.
"[Clayton Keller] likes to make plays, that line likes to make plays," Kesselring said. "I kind of saw that he saw me, so I got on my horse a bit and he made a great pass. A little alley-oop and luckily it went in."