James Neal put the Blues ahead 4-3 with a backhand at 11:42 of the second period, and Robert Thomas scored into an empty net at 19:00 of the third period for the 5-3 final.
"Going into the third period with a lead, you want to lock it down as much as you can," Neal said. "It's a good team win, really liked the way the guys battled."
Thomas had a goal and an assist, Tyler Bozak had two assists, and Joel Hofer made 23 saves and had an assist in his NHL debut for St. Louis (7-1-1).
"There [were] obviously a lot of emotions going through my head," Hofer said. "It's obviously super special. You dream of it your whole life, and to get the win, in the first game, too, it's the cherry on top."