Batherson pushed it to 5-0 at 12:35 with a one-timer off a cross-slot pass from DeBrincat, and Stutzle made it 6-0 at 7:51 of the third period by scoring five-hole on Greiss on another breakaway.
"I think our group is more mature now that they're letting games go," Smith said of his players rebounding after a 4-3 overtime loss to the Chicago Blackhawks on Friday. "I think people on the outside made so much noise about losing to Chicago, but any team can beat any team. We're witness to that. … I think our guys just knew we didn't play the right way.
"They turned the page, which is a good sign because if you win a game, you're not going to be too high either. And I think that's where the leadership has gotten a lot better that way."
Parayko made it 6-1 at 9:16 with a backhand from the slot before Pitlick tipped a point shot from Justin Faulk to make it 6-2 at 11:06.
''We've got to move past it at some point, right?" Pitlick said regarding the trades. "They were big pieces in our locker room and on the ice, but I think we have the right group, and the guys here that can look past it and move past it."
Ridly Greig scored his first NHL goal at 18:16 for the 7-2 final.
"I'm not much of a celebrator," Greig said of his subdued reaction. "Maybe if it was a 2-2 game, I probably would have done something, but it was a good feeling."
NOTE: St. Louis forward Jake Neighbours sustained an upper-body injury in the first period and did not return. "He's probably out a while," Berube said. "I'm not sure."