Midfielder Marshall struck the decisive goal six minutes after half-time as he drifted in from the left to rifle a shot in off the post, and the visitors were given an uphill task on the hour as Neal Ardley saw red for a second booking.
Both sides had chances, with Birmingham’s Clayton Donaldson prodding wide early on and Paul Caddis denied by Jason Steele.
After the goal and sending-off, Blackburn dominated, with Colin Doyle producing wonderful saves to deny Tom Cairney and Markus Olsson.
Cairney also struck a post but Birmingham almost snatched a draw with a superb Andrew Shinnie effort that clattered the crossbar.
The win halted a Rovers run of four game without victory, while Birmingham, who parted company with manager Lee Clark on Monday, have now won just one of their last 11 league games.
Home boss Gary Bowyer brought in Alex Baptiste for suspended defender Matt Kilgallon, while Doyle was in goal for the visitors following Darren Randolph’s red card against Bolton.
Crosby, who guided Sunderland to the 1992 FA Cup final, picked the Birmingham team.
Corry Evans should have put Blackburn ahead early on but somehow failed to connect with a Jordan Rhodes cross, and Birmingham missed a great chance to punish them as Donaldson poked a shot wide when sent clean through.
Koby Arthur forced Steele into a low save after 27 minutes and the goalkeeper produced a superb fingertip save three minutes later to prevent Caddis’ thunderous strike finding the net.
They were made to pay after 51 minutes as Rovers took the lead when Marshall cut in from the left and struck home his fifth of the season.
But Steele almost gifted Birmingham an equaliser by flapping at a cross which allowed Caddis to fire a shot from inside the area that the goalkeeper blocked.
Blackburn were given a boost though on the hour as Eardley was sent off for a second booking when bringing down Olsson who threatened to burst clear.
It was left to Doyle to keep Rovers at bay after 64 minutes, first denying Marshall with an outstretched leg and as Cairney seemed certain to score from the rebound, he flung himself to the left and parried.
He produced another flying save to deny Olsson four minutes later before Wes Thomas stung the palms of Steele.
Birmingham’s goal continued to lead a charmed life as Marshall set up Cairney but his shot took a deflection and struck the base of the post with 10 minutes remaining.
Birmingham were inches away from grabbing an equaliser in injury time but Shinnie’s rasping 25-yard effort cannoned back off the crossbar with Steele beaten and Rovers held on.