Greaves stopped 17 of 18 shots in the third period to hold off a late push by the Blues.
“It was a bit busier but I think that's what you play for, is those moments,” he said. “The guys did a good job of clearing the second chances and everything, so it made my job a little easier.
“It's fun for me. I'm just trying to focus on what I can do to help the guys win the game. Just try to stop each puck that comes my way.”
Zach Werenski scored for the third straight game and an assist, and Charlie Coyle and Sean Monahan each had goals for the Blue Jackets (7-4-0), who have won six of seven.
“It took everyone from our penalty kill to our power play, Jet, everyone throughout our lineup,” Werenski said. “It's a big win for us. I feel like we can maybe be a little bit more aggressive in the third when we have leads. I felt like we were on our heels a little bit there, but at the end of the day, we got the job done.”
Justin Faulk and Nick Bjugstad scored, and Joel Hofer made 32 saves for the Blues (3-7-2), who have lost seven straight (0-5-2).
“What I liked was our defensive habits were significantly better,” Blues coach Jim Montgomery said. “And I liked our first period and our third period.
“The biggest dislike from the game was too many turnovers, trying to go east-west against a team that had three back all night long. In the third period, we got in behind them and worked and forechecked and ended up generating a lot of scoring chances.
“We're trying to stay in the moment. That's the way we always are with our process. And right now, the last two games, our process has been good and as we continue to improve upon it then the results will come.”
Faulk put St. Louis up 1-0 at 3:37 of the first period. Oskar Sundqvist sent a cross-ice pass from the left boards to Faulk in the right face-off circle, who beat Greaves with a wrist shot.
Coyle evened it 1-1 at 4:38 by jabbing a loose puck in the crease over the goal line off the right skate of Faulk. Coyle also had four assists against the Toronto Maple Leafs on Wednesday.