By then they were down to 10 men after Jordi Lopez received a second yellow card, and then in the first minute of injury time Nathan Dyer was also dismissed.
"I am very proud of my players and their attitude against a very strong Sheffield United side," said Swansea manager Paulo Sousa.
"I want more than the strikers to score and Tom Butler has certainly stepped up after being out through injury for six or seven weeks.
"The most important thing however, is that the players believe in what we are trying to do here.
"I am happy that important players like Ferrie Bodde are back on the field but it is too early to say the extend of the injury he received today.
"We will be short on players through both injury and suspensions next week but we will see who is available and go from there."
The shot-shy Swans had been without a goal in over six hours of football and were thrown a lifeline just after the break when Nick Montgomery brought down Trundle.
Trundle lashed in the spot-kick to give his side the lead.
The Blades stormed back and Darius Henderson was denied a goal for offside before Quinn equalised with a cheeky backheel.
Butler, who had entered the fray on the hour, combined with Dyer before rifling home a 20-yard shot which whistled into the corner of the net.
There was still time for Dyer to be dismissed but the nine men hung on for Sousa's first home victory since taking control.
Blades boss Kevin Blackwell felt that his side had thrown away the points.
"We controlled the game but not the result. We gave away two sloppy goals and after that it was always hard to get a result," he said.
"We started well but had to dig in a bit, although we went in at half-time full of confidence.
"We had a good goal disallowed but after equalising there was only one side in it.
"Two breakaways and credit to Swansea as they took the only two chances they had in the second half.
"I am not complaining about the effort but there is a lot for us to learn from today."