Marshall was a late withdrawal for the Dragons due to a hamstring strain he picked up in the Cowboys match, and his teammates were unable to ignite their attack in his absence.
It is Brisbane's fifth win of the year, third in a row and puts them clear on top of the ladder above second-placed Melbourne, who host Canterbury on Monday.
Coming off a tough five-day turnaround, the Dragons completed just nine of their 17 sets in the first half and were fortunate to go into the sheds only 12-0 down.
Corey Oates opened the scoring for Brisbane just three minutes in as the recipient of a mesmerising cut-out ball from the in-form Anthony Milford.A Corey Parker penalty soon afterwards gave Brisbane a 6-0 lead, and then Alex Glenn slammed down a grubber from Ben Hunt in the 29th minute to widen the margin.
Even when the Dragons did generate openings, they were discovering brand new methods of self-sabotage.
Lock Jack de Belin looked certain to end a two-week wait for a St George Illawarra try when he brought two Brisbane defenders with him over the line in the 18th minute.
But replays showed his own arm was stopping him from grounding the ball.Then early in the second half, winger Peter Mata'utia was played into a yawning gap just three metres out from the line - only to infuriate Dragons fans further by knocking on.
The Broncos duly made them pay as heavy legs began taking over late in the match, with Milford scoring in spectacular style right on the dead ball line in the 72nd minute, before a Hunt flick pass unleashed Jordan Kahu two minutes later.
Coach Wayne Bennett did not appear to overly thrilled by Brisbane's performance.Asked what he liked about their display after the match, Bennett said: "Two points - that's it."I just didn't like the way we played tonight. It was just a disjointed game.
"It was a better second half, a little bit, but the first half never really happened for either team."
Dragons coach Paul McGregor said the effort of his players couldn't be questioned.
"(Brisbane) did score 14 points in the last 15 minutes, but that was on the back end of a five-day turnaround and a North Queensland trip and they are a quality side," he said.
"I'm comfortable with their effort and commitment to it."It's just we've got to be a lot more intelligent and a lot more clinical."