Rasmussen made it 2-0 at 4:21, finishing a 2-on-1 rush with Lucas Raymond.
Robby Fabbri then extended the lead to 3-0 at 5:24 when Kasperi Kapanen's clearing attempt deflected in off him from the edge of the crease.
"It's not that we didn't come out ready," Blues captain Brayden Schenn said. "We just weren't very smart in the first five minutes."
DeBrincat made it 4-0 at 19:02 with a power-play goal.
"We knew we hadn't been getting the starts we needed," DeBrincat said. "That was a point of emphasis tonight, and we came out with three quick goals. After that, we just wanted to keep it going."
Bolduc got the Blues to within 4-1 at 12:28 of the second period, scoring on his 21st birthday in his second NHL game.
"I'm pretty lucky, I didn't have to wait long for my first one," he said. "It felt good, but this was an important game and a tough loss."