Zach Sanford scored 36 seconds into the second period to extend the lead to 4-1, and Robby Fabbri made it 5-1 at 11:53 and end Saros' night.
"It ended up being good timing early in the second period to get that lead up to three goals instead of two," Sanford said. "We've seen a couple two-goal leads disappear. To show that we were going to keep coming was really huge for us."
Pietrangelo scored at 14:50 of the third period to make it 6-1 before Hartman cut the lead to 6-2 at 16:13.
"We should be this team the way we played tonight," Blues forward Pat Maroon said. "This is what we should do every single night. There's no reason why we couldn't do that every night, controlling the puck, playing good defensively, closing hard, physical, going to the net, being a frustrating team to play against."