Dvorak scored twice in a 40 second span in the third period.
He knocked in a puck at the side of the net following a Blues turnover 41 seconds into the period to increase the lead to 5-3. He then made it 6-3 at 1:21 when he tipped in a wrist shot from Josh Anderson from the blue line.
"We faced a little bit of adversity down two in the second and we fought our way back," Dvorak said. "The power play got us going and we kind of took momentum of the game there, and we had a good comeback for us on the road."
Tarasenko cut it to 6-4 at 2:47 following a Montreal turnover, but Dvorak completed the hat trick with an empty-net goal at 18:25 for the 7-4 final.
Kyrou gave the Blues a 1-0 lead at 7:47 of the first period when Schenn's shot from just inside the left circle caromed off his skate and trickled through Allen's pads.
Suzuki tied it 1-1 at 15:08, tipping in a shot from Jordan Harris after Blues defenseman Nick Leddy turned the puck over in the zone.
"You don't need to make [that play]," St. Louis coach Craig Berube said. "We've got to simplify, for sure. He could've went up the wall with it and been out of the zone. We're not simple right now. We complicate things. It was a good first period, I thought, and from then on, like in the second, we've got complete control in my opinion and then we let it slip away."
Noel Acciari's first goal with the Blues, a rebound of a Tyler Pitlick shot, gave them a 2-1 lead at 5:10, and Kyrou's power-play goal at 9:38 made it 3-1.