Garry Monk, Gorka Pintado and Angel Rangel were all dismissed by overworked referee James Linnington as the visitors won 2-1 in extra-time.
With second-half substitute Stephen Dobbie already off the field injured it left Swansea with just seven players and when Besian Idrizaj also went off the field with a few minutes remaining it looked as though the referee would be forced to abandon the match as Swansea only had six fit players on the field.
Idrizaj was persuaded to go back onto the field and the game finished a few minutes later to a chorus of boos from the unhappy home crowd.
Swansea manager Paulo Sousa would not speak about the referee's performance.
"We have been provoked and we have reacted not in a good way but my way is to protect my players.
"We have our opportunity to write to the FA about the referee's performance but at this moment we do not know what offences the players have committed.
"I never wanted the match abandoned but Idrizaj should not have gone back on the field but as we were then down to only six players the referee put pressure on us to get him back on."
It was the visitors who took an early lead when Niall Canavan headed home a Josh Wright corner in the 14th minute.
Sousa was forced to bring on his heavy guns during the second half and it paid off when Dobbie equalised with a rasping drive in the 78th minute.
In the 97th minute Dobbie received a kick in the stomach and was forced to leave the field to be followed 10 minutes later by Pintado who received a second yellow card for a lunging tackle.
Two minutes later Alan Tate handled a Paul Hayes goal-bound shot inside the box and Gary Hooper slotted home the spot kick to give his side the lead.
Rangel followed Pintado and Monk to the dressing room and then came the farcical finish as Idrisaj was forced back on so that the game could end.
"It is the first time in 21 years that we have reached the next round of the Carling Cup," said Scunthorpe manager Nigel Adkins.
"Swansea's lack of discipline has cost them dearly as they are an excellent side. They are a fantastic passing team and we are really pleased that we defended well against them.
"The discipline of my players was also very pleasing and the referee had no choice but to send their players off as they seemed to lose their composure."