The Bluebirds' winner came on the half-hour through Paul Parry after a clever reverse pass from the impressive Peter Whittingham and they hung on for all three points.
The Blades should have been ahead long before that but former England striker James Beattie missed a couple of glorious chances.
The first came straight from the opening whistle when Beattie blasted over from close range.
He then missed with a header from a Keith Gillespie corner after Parry had failed to clear.
"I thought we started well but we missed a few chances and James Beattie should have put us in front in our first attack," said a disappointed Robson.
"You cannot afford to miss chances like that at this level. Once Cardiff went ahead they defended very well to keep us out and our performance particularly in the second half became very scrappy.
"We have put teams under pressure in the past after going behind but Cardiff were excellent in defence and they have some very good players".
United were dealt a blow in the 58th minute when their skipper Chris Morgan limped off to be replaced by one-time City target Chris Lucketti.
Two minutes later Robson went for broke by introducing Rob Hulse and Lee Hendrie in an effort to get back on level terms.
Once City were ahead they began to play with confidence and both Whittingham and Parry went close with shots that had Paddy Kenny leaping across his goal.
Following the goal Whittingham powered in three superb efforts that Kenny somehow kept out.
With four minutes remaining the visitors were awarded a free-kick just outside the area after Michael Tonge was upended but Phil Bardsley's rasping drive flew past the right-hand post.
In the closing minutes Cardiff were able to break out of defence and substitute Steve MacLean and Whittingham both went close to sealing victory.
Cardiff boss Dave Jones said: "I was very pleased as from start to finish it was a good performance from all our players.
"We said at half-time that a clean sheet would give us the win and that we had to cut out the supply to their front men and we did just that.
"The goal was very well worked and Peter Whittingham's reverse pass to Paul Parry for the goal was superb.
"When you look at the division now we are a match for anybody in the league and today we carried on from last week's second half performance at Bristol.
"We have come in for a lot of flak recently but today we deserve praise for an excellent performance. The three points today were important because the other sides around us have also won."
The news that Chris Gunter was on his way to Tottenham in a big money move had dismayed home fans prior to the match and it is only hoped that no more of Cardiff's young stars leave in the January sales.
"Everyone is disappointed that Chris is going but we are delighted he is joining a team like Spurs and we wish him all the best" said Jones.
PA Sport