The Blues were glad Schwartz returned. His third-career NHL hat trick was his first since April 3, 2015 against the Dallas Stars and three years to the date of his first, Oct. 18, 2014, against the Arizona Coyotes.
"When he's going, he brings everyone else into it," Allen said of Schwartz. "He's really the guy that gets guys going, sparks guys, gets guys the puck. He works his best down below the goal line. ... He's really coming into a league of his own."
The Blackhawks (4-2-1) got power-play goals from Richard Panik and Ryan Hartman, and Corey Crawford made 28 saves.
"They played well and we were brutal," Blackhawks coach Joel Quenneville said. "It was a bad start, it was a bad middle, it wasn't very good. Even a little excited at the end, but that wasn't very good. That was as close to brutal as you can get."
Schwartz's goal at 4:46 of the first period gave him a point in six of seven games this season.
Schwartz made it 2-0 at 5:59 of the second period. Tarasenko, who also has a point in six of seven games, scored from the slot at 7:49 to make it 3-0.
Brodziak scored off a 2 on 1 with Tarasenko at 7:01 of the third to make it 4-0.
The Blackhawks ended Allen's shutout bid on Panik's goal with 5:13 remaining, and Hartman made it 4-2 with 3:53 remaining that needed video review after the Blues challenged for goalie interference.
Schwartz got the hat trick when he scored into an empty net with 51.1 seconds remaining.