Benin international striker Gestede has been heavily linked with a move away from Ewood Park in recent days, but popped up in the 78th minute to put the home side in front before Conway netted with a low drive 11 minutes later.
Kyle Bartley had seen red in the sixth minute after being judged to have brought down the lively Josh King just outside the penalty area as the last man, but it was the visitors who broke the deadlock.
The match – played out at a sparsely populated Ewood Park – looked destined for a replay during a quiet second half, but Gestede's 13th goal of the season put Championship side Blackburn ahead before Conway added a third.
Swansea's bad day at the office got worse when Sigurdsson was sent off for a late tackle on Taylor in the closing stages.
The Welsh side have been linked with a move for Blackburn's Jordan Rhodes in recent days, and the Scotland international, along with fellow in-demand frontman Gestede, had to settle for a place on the bench, but lone striker King soon proved his worth by making the run that led to Bartley's early dismissal.
However, it was the 10 men of Swansea who broke the deadlock in the 21st minute – Sigurdsson burying a wonderful strike from 25 yards after drifting in from the left.
Marcus Olsson's cross from the left was nodded down by Cairney, with Taylor turning well on the edge of the six-yard box to power home and leave Fabianski with no chance.
Conway – who like Gestede played for Swansea's fierce rivals Cardiff City – made absolutely sure of the win with a low drive that squirmed past Fabianski.
Sigurdsson's red then completed the visitors' misery after a late challenge on Taylor.
