Crouse tied it 1-1 at 15:44 with a power-play goal, coming off the bench and taking Clayton Keller's pass and beating Hofer with a wrist shot from the right circle.
"I thought we did a good job of responding and not getting overwhelmed when things weren't going our way," St. Louis defenseman Torey Krug said. "That's the sign of a team that's growing up a little bit. We've got to continue that. There's always that next play to be made, there's always going to be another power play when you do make those mistakes. We've got to continue with that."
The Coyotes were outshot 15-3 in the first period after killing off four minor penalties, including a full two minutes of a 5-on-3.
"They give us a chance to win that game," Tourigny said of Arizona’s penalty kill. "They give a chance to be in the game. They were rock solid."
St. Louis went 0-for-7 on the power play and is an NHL-worst 1-for-35 (2.9 percent).
"Obviously you want to be better, you want to help the team out, but at the end of the day, it's a big win," Thomas said. "Those guys have had our number lately, especially coming in here. I think we did a great job of limiting their offense, playing a smart game, especially in the third period."
NOTES: Hofer is 3-0-0 with a 1.34 goals-against average and .950 save percentage in his past three starts after allowing six goals on 42 shots in a 6-2 loss against the Coyotes on Oct. 19. ... Thomas, who has a five-game point streak (five goals, one assist) also has 10 points (six goals, four assists) over the past nine games. ... St. Louis defenseman Scott Perunovich played 14:51 and had three blocked shots in his first game since Oct. 27. ... Crouse has eight points (five goals, three assists) in his past seven games. ... Coyotes forward Matias Maccelli’s nine-game point streak ended (two goals, eight assists) in his 100th NHL game. ... Arizona forward Nick Bjugstad's seven-game point streak ended (one goal, six assists).