Frenchman Rudy Gestede fired home at the second time of asking in the 78th minute to give Championship (second tier) outfit Blackburn a 2-1 lead before Craig Conway's powerful shot squirmed through Lukasz Fabianski's hands after 87 minutes.
Swansea defender Kyle Bartley was shown a straight red card for bringing down pacy Blackburn forward Josh King in the seventh minute but Gylfi Sigurdsson's long-range rocket put the top flight side ahead shortly afterwards.
Chris Taylor volleyed in to restore parity one minute later.
Swansea's frustration boiled over in stoppage time when Sigurdsson was shown a straight red card for needlessly scything down Taylor.
In other fourth-round ties on Saturday Chelsea host third-tier Bradford City, Manchester City welcome second-tier Middlesbrough, Tottenham Hotspur entertain Leicester City and Southampton play Crystal Palace.
(Reporting By Tom Hayward; editing by Martyn Herman)
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