Former Manchester United captain Robson saw his side clinch a place in the fifth-round with a 2-1 win over Manchester City after holding onto a first-half lead courtesy of goals from Luton Shelton and Jon Stead.
Rising teenage star Daniel Sturridge gave City hope with a stunning volley three minutes after his arrival as a half-time substitute, but United were barely troubled beyond that.
• Robson: We owed Blades fans
For Robson, the victory will have come as a major relief as there had been rumours this week his job was on the line following last weekend's 2-0 derby defeat at Sheffield Wednesday.
But if there was any suggestion his team did not want to play for him, that was by blown away by this performance, and the first half in particular.
It will have had most United fans wondering just why their team has performed so miserably on occasions in the Coca-Cola Championship this season.
But after the opening 45 minutes Robson's side were rewarded with a rousing reception as they left the pitch as they led 2-0 against a strong City side flying high in the Premier League.
Even Robson must have chuckled to himself, though, at the comical nature of his side's first goal inside 12 minutes.
A swift break down the left wing resulted in Lee Martin - on loan from Manchester United - delivering a low, piercing cross into the heart of the area.
The ball initially took a slight deflection off the leg of City captain Richard Dunne, and then onto one of the myriad of blue balloons floating around the area courtesy of the 6,000 City fans behind Hart's goal.
Michael Ball was clearly confused as he completely missed his attempt at a clearance, while Shelton was almost also deceived.
But the Jamaica international managed to steady himself and slip a simple six-yard shot past Hart for only his third goal for the club following a £1.85million move from Helsingborgs a year ago.
Emile Mpenza planted a free header over the crossbar from eight yards for City, then Elano struck the left-hand post with a right-foot shot from 20 yards, and followed up shortly after with a curling free-kick that kicked up off the turf, only for Paddy Kenny to pull off a sensational one-handed save.
Against the run of play, the Blades made it 2-0 to the unbridled joy of Robson on the touchline.
This time the goal stemmed from the right wing and a cross from Derek Geary the City defence struggled to clear, with the loose ball falling for Stead to drive home from 12 yards.
United managed to see out the remainder of the half without too much threat from their opponents, only to be caught out by a slice of magic from City's up-and-coming striker Sturridge.
The 18-year-old replaced Elano at half-time and within three minutes of his first appearance of the season he was on the scoresheet.
Eriksson raved about the youngster - nephew of former Derby and Wolves striker Dean - in Friday's pre-match press conference, and it was easy to see why.
Sturridge, who scored a hat-trick in City's 3-1 FA Youth Cup win over Reading on Wednesday, was in the right place to take a ball that had skimmed off the head of a falling Dunne from a Corluka corner.
Showing the composure of a veteran, he then knocked the ball up with his right knee before volleying home from 15 yards with his left foot.
That should have been the cue for City to mount a comeback, but not once did they lamentably manage to test Kenny again, allowing United to take their place in the last 16 with ease.
Robson: We owed Blades fansSheffield United manager Bryan Robson believes the 2-1 victory over Premier
League high-flyers Manchester City in the FA Cup fourth-round tie at Bramall
Lane was the ideal tonic after a derby defeat at Sheffield Wednesday last
weekend.
'I thought we were terrific and we owed the fans that one,' Robson said.
'We have been missing far too many chances but today they went in for us.
'I thought we matched them in every department and that is what the lads are
capable of.'
Robson admitted their first goal had a touch of luck about it but was happy to
accept it.
'There was a bit of fortune about it. Today we did ride our luck at certain
times but our counter-attack play was very good,' he told Sky Sports.
Manchester City manager Sven-Goran Eriksson blamed referee Alan Wiley for not
preventing the circumstances which led to the opening goal.
'It is a very strange goal because the ball changed direction. Two balloons
made a one-two with Michael Ball. I have never seen a goal like that,' he
said.
'We asked the fourth official twice in the first half to tell the ref to stop
it (the match) and clear it.
'I think the ref told Joe Hart (City's goalkeeper) that when the ball is in
the other half of the pitch he should clear it but I think that is not right -
you should not ask a player when the ball is live to clear the pitch.
'But it would be stupid of us to say it (the defeat) was because of the
balloons.'
Eriksson was also critical of his players for lacking the stomach for a
battle.
'I think we were not happy to take the fight as we should have done. We lost
too many balls in midfield and up front,' he said.
'It is the beauty of the cup which hit us today. If you are not prepared to
battle, as well as play good football, you know a team from a lower division
will fight well.
'We should do better. It is our fault in the first half, although second half
we did well.'
Sheffield United 2 - 1 Manchester City
