The Premier League side had Kyle Bartley dismissed after a sixth-minute coming together with the speedy Josh King, but took the lead 20 minutes in thanks to a thunderous Gylfi Sigurdsson long-range effort.
Blackburn 3-1 Swansea
The FA Cup
Saturday 24 January 2015
Winners receive £90,000
Chris Taylor volleyed the hosts level just a minute later however, and their numerical advantage took its toll over 85 minutes, with Gestede putting them ahead with ten minutes remaining before Conway put the icing on the cake with an opportunistic late long-ranger.
Sigurdsson saw red in injury-time for kicking out late on.
Blackburn Rovers boss Gary Bowyer said: “I am delighted because we knew it would be tough. But Bartley’s early sending-off made it into a contest.
"It was a harsh sending-off, but if the referee says it is a foul, he has to follow the letter of the law.
“Their opener was a wonderful strike from a wonderful player, but we showed great character to respond so quickly.
"Taylor has scored a few in the cup and sometimes players can go on a run of doing well in the FA Cup.
Watch the goals from Saturday's tie at Ewood Park
“I told Gestede to go on and score the winner. He’s ex-Cardiff and I said just to go and do your stuff.
“We have just had a blip of form recently, but the players are sticking together through it. We just need the supporters to stay with us; sometimes it can take more than 18 months for things to change around."
Swansea manager Garry Monk added: “I am disappointed to lose of course – I wouldn’t say they were the better side, but ten men makes it more difficult.
"I didn’t think they were a threat in second half and wanted to push for a goal on the counter in the final ten.
“We should have dealt with the threat a lot more for those final two goals though.”
Blackburn Rovers (4-5-1): 13 Simon Eastwood; 2 Grant Henley, 22 Shane Duffy, 4 Matthew Kilgallon, 14 Markus Olsson; 19 Chris Taylor, 17 Lee Williamson, 6 Jason Lowe, 32 Craig Conway, 10 Tom Cairney; 7 Josh King.
Substitutes: 8 David Dunn for Williamson 41, 39 Rudy Gestede for King 60.
Substitutes not used: 5 Grant Hanley, 9 Chris Brown, 11 Jordan Rhodes, 12 Ben Marshall, 30 Jason Steele.
Goals: Taylor 23, Gestede 78, Conway 89.
Swansea City (4-5-1): 1 Lukasz Fabianski; 22 Angel Rangel, 2 Jordi Amat, 27 Kyle Bartley, 19 Dwight Tiendalli; 14 Tom Carroll, 8 Jonjo Shelvey, 12 Nathan Dyer, 23 Gylfi Sigurdsson, 58 Modou Barrow; 18 Bafetimbi Gomis.
Substitutes: 33 Federico Fernandez for Carroll 9, 20 Jefferson Montero for Barrow 62, 17 Miguel Nelson Oliviera for Gomis 77
Goals: Sigurdsson 21
Referee: Craig Pawson