Logan Brown had a goal and an assist for the Blues (48-20-11), who are 13-0-2 during the streak, a team record. Jordan Binnington made 16 saves.
St. Louis moved into a tie for second in the Central Division with the Minnesota Wild, who have one game in hand.
"I think we might have just gotten away from it a little bit, but you learn from it and at the end of the day, we came back and we again got two points," Brown said.
Bokondji Imama, Michael Carcone and J.J. Moser scored in the third period for the Coyotes (22-50-7), who have lost 10 straight (0-8-2). Harri Sateri made 28 saves.
"They're a really good team and we had a tough time creating offense, but we found a way and found our rhythm and had more cohesion," Arizona coach Andre Tourigny said. "There's a lot of positive there, a lot to build around in that game."
Imama, who was playing in his second NHL game, cut it to 4-2 at 5:39 of the third off a backhand pass from Alex Galchenyuk near the top of the crease.
"'Boko' is a longtime minor league player who made his way, fought, fought through adversity, and he made it," Tourigny said. "It's a little bit like Carcone."