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.
Gylfi Sigurdsson pulled the trigger from distance to open the scoring with a stunning strike after 21 minutes, although that lead was soon cancelled out by Chris Taylor.
The Blackburn midfielder turned in the area after some neat build-up play to power the ball beyond goalkeeper Lukasz Fabianski two minutes later.
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.
The centre-back had a coming together with King just outside the box and duly was given his marching orders.
Fabianski got down well to keep out the resulting free-kick from Tom Cairney, who tested the goalkeeper again with a curling effort from distance after 13 minutes.
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.
It took just two minutes for Blackburn to pull level, though.
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.
Cairney deceived Jonjo Shelvey with a clever turn in the box before firing over at the beginning of the second half, as he and King continued to cause problems for the Swansea back line.
Gestede came on in place of the impressive King on the hour mark, rendering him cup-tied should he leave Ewood Park, while Swansea were able to bring on Jefferson Montero after a month out with a hamstring injury.
Blackburn boss Gary Bowyer also brought on Rhodes, but it was Gestede who gave the hosts the lead - eventually prodding home from close range after Swansea failed to clear their lines from a left-wing cross.
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.
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