Jordan Kyrou scored, and Ryan Suter had two assists for the Blues (6-6-0), who ended a three-game skid. Jordan Binnington made 35 saves for the Blues.
"We had an OK first, but I think in the second period, we just got pucks deep," Parayko said. "Had good line changes, I thought, where we got pucks into their zone and we would just get into the O-zone and just roll them over and keep going, and I think took over the game. It was good, I thought. A good game."
St. Louis defenseman Philip Broberg left the game because of a lower-body injury at 7:02 of the second period when Toronto forward Mitch Marner fell on his right leg along the left boards in the offensive zone. He skated off on his own but limped down the tunnel.
Blues coach Drew Bannister had no update after the game and said Broberg would be reevaluated on Sunday.
"He's been battling so hard for us and taking hits," Binnington said of Broberg. "He's a warrior, though, I know. He'll be back as best as he can, when he can. It was good to see guys step up and pick each other up. It was a tough play."
Marner and Steven Lorentz scored for the Maple Leafs (6-5-1), who had won their past two games. Joseph Woll made 20 saves.
"For me, it's a mindset. We didn't play in the second the way we played in the first," Berube said. "Direct, hard on things, pucks. I thought that we were light in the second period."