Jaskin had two assists, and Brodziak had a goal and an assist to help St. Louis (9-2-1) extend its point streak to six games (5-0-1) and remain undefeated at home (4-0-0).
Jake Allen made 36 saves, defenseman Joel Edmundson had a goal and an assist, and Vladimir Sobotka scored for the Blues, who have won 11 in a row in the second game of a back-to-back dating to last season.
Matt Calvert scored for the Blue Jackets (7-4-0), who lost for the first time in three games. Joonas Korpisalo made 38 saves.
"Both teams came in on a back-to-back," Columbus coach John Tortorella said. "They played better early on and grabbed momentum right away, and they get the lead. It's tough against a veteran team like that in your back end. It's tough to come back when you're down by a couple.
"I felt we put in some good minutes. I thought we didn't have a lot of puck luck where I thought we were getting closer. Allen made some key saves, some of them I don't even think he saw the puck, but the team that won tonight deserved to win."