There were three changes in the City starting line-up with Gavin Rae, Mark Kennedy and Michael Chopra all in from the off.
Cardiff born Wales international David Cotterill was in the Blades line-up as the visitors looked to extend their five-month unbeaten away record.
It was Cotterill in the fourth minute who almost caught City keeper Stuart Taylor napping with a centre from the left that went right across goal and struck the far upright before being cleared.
Disaster struck City in the 24th minute when Darius Henderson chased a long through ball from Greg Halford and when Gabor Gyepes moved in to tackle, Henderson went down in the area.
Referee Paul Taylor immediately issued a red card and signalled a United penalty which Cotterill rammed home to put City behind and down to ten men.
Dave Jones immediately brought Chopra off and replaced him with defender Miguel Comminges, pushing Kevin McNaughton into a central position.
On the half-hour, Cardiff made another change when Stephen McPhail went on for Chris Burke but they were in even more trouble four minutes later when Ross McCormack became the second City player to be dismissed.
He tangled with Kyle Naughton and the referee had not hesitation in showing him a red card for an alleged elbow to reduce the Bluebirds to nine players.
As an explosive first half drew to a close, Rae had a header cleared off the line in an isolated attack but then Taylor had to spread himself to push away a Naughton shot just as the referee signalled the interval.
In the first minute of the second period, United doubled their lead when slack marking by the home defence left substitute Jamie Ward unmarked to head in from a corner.
When Comminges left the field for treatment to a head injury, City played for a couple of minutes with eight men but fortunately he was able to resume.
With City tiring, Eddie Johnson was sent on for the final four minutes in place of Jay Bothroyd, who had worked tirelessly up front on his own for most of the game.
Three minutes from time, Stephen Quinn made it an emphatic scoreline for the Blades when he ran through to shoot across Taylor and into the far corner of the net.