The visitors’ task was made all the more difficult after just seven minutes as Kyle Bartley was given his marching orders for a foul on Joshua King, with the Swans defender adjudged to be the last man.
However against the run of play, Swansea took the lead thanks to a fantastic strike from Icelandic midfielder Gylfi Sigurdsson from fully 25-yards.
The lead lasted just two minutes as Chris Taylor beat Lukasz Fabianski to equalise after good work from Tom Cairney.
Neither side created much in the opening exchanges of the second period until Rovers’ boss Gary Bowyer brought on his dynamic strike pairing of Rudy Gestede and Jordan Rhodes, and minutes after the latter had entered the fray, Gestede had put the hosts in front.
Swansea rallied, but they could not find an equaliser, and they were caught on the break as Craig Conway netted a third past the despairing dive of Fabianski.
The south Wales side’s misery was compounded when Sigurdsson was sent-off late on, but in front of a surprisingly low crowd at Ewood Park, Blackburn’s strikers once again proved their worth.
Despite leaving Gestede and Rhodes on the bench, Blackburn were in the ascendancy from the off, and their task was made easier inisde the opening ten minutes.
King, playing in the lone striker’s role, raced onto a hopeful long ball, was too quick for Bartley, causing the Swans defender to bundle the pacey Rovers forward to the floor.
Referee Craig Pawson had no option but to brandish the red card, and left Garry Monk with a real headache.
However, the visitors looked anything but hindered by the dismissal, and took the lead 15 minutes later.
Sigurdsson looked to be too far out for a strike at goal, but he unleashed a ferocious effort that Simon Eastwood had no chance of keeping out in the Blackburn goal.
Just two minutes later, the home side restored parity with another goal of real quality.
A long ball to the back post picked out Cairney, who got up brilliantly to head the ball back across goal and, on the turn, Taylor hooked the ball past a helpless Fabianski to level proceedings.
King was causing the Swansea backline all sorts of problems with his pace, and but for several last ditch blocks, the Norway international could have put the home side in front before the break.
After the interval Bafetimbi Gomis broke clear almost immediately, cut inside but saw his powerful shot brilliantly blocked by Matthew Kilgallon.
The hosts were doing all the probing, but they could not create that killer chance, with Swansea looking comfortable at the back.
However, the introduction of Gestede and Rhodes changed things, and the former notched what proved to be the decisive goal in typical goal poaching fashion.
A cross into the box was not dealt with by the Swansea rearguard, and Gestede pounced onto the loose ball to prod home his 13th goal of the season, and further alert potential Premier League suitors to his talents.
Swansea threw men forward as they went in search of a leveller, but on the break Conway darted forward and fired past Fabianski to make sure of the win and a place in the last 16.
The Polish stopper will be unhappy though, as he really should have kept it out after getting two hands to the strike.
Sigurdsson was given his marching orders for a late lunge on Taylor to make sure of a miserable journey back to south Wales for the Swans faithful.