it is probably easier just to get the prosaic stuff out of the way first: Manchester United yesterday progressed to the fourth round of the FA Cup, where they will meet Liverpool, with the tie to take place at Anfield.

It is telling, though, that the Old Trafford club will be less concerned about that fact than they are with the outcome of their third-round win over Manchester City. Which is understandable, given it is all anyone else is going to be talking about.

United had to win yesterday. Having suffered defeat in their last two matches – meetings with Blackburn Rovers and Newcastle United in the Barclays Premier League – a few aspersions had been cast about their ability to maintain their authority over City, and their designs on the league title. Was Sir Alex Ferguson too old? His squad too young? Is Wayne Rooney looking for a way out of the club? Again.

The latter seemed to be answered emphatically, as if a hastily arranged statement on Friday night wasn't enough, with the United striker scoring twice in the first half, making sure to kiss the badge on his shirt so as to confirm his loyalty to the club.

It was a little trite but it seemed to have the desired effect. Rooney's first goal came after just 10 minutes, the England internationalist flicking in a header off the underside of the crossbar from Antonio Valencia's cross. His second was a rather more sheepish nod over the line just before half-time after his penalty had been saved by Costel Pantilimon, the City goalkeeper.

"We have seen the good stuff today but any flaws will be absolutely annihilated by the press," said Ferguson, who had been irked by rumours of a falling-out with his striker. "That is what they are like. That is what we are dealing with."

Rooney's double – which sandwiched an adroit volley from Danny Welbeck – will have felt like acts of defiance against the rumours and criticism which have stalked United this season. Ironically, much of that stems from their meetings with City; there was the Community Shield in May – a match United won but which shattered their belief that City were not a formidable title challenger – and United's 6-1 loss at Old Trafford which so wounded them.

This latest meeting might have proved cathartic, then, for United had it not been for the second half. Ahead by three goals and having rediscovered an authoritative swagger to their play, the visitors were expected to crush their rivals. They did not.

Instead what transpired was a fraught 45 minutes which left United clinging to their lead, goals from Aleksandar Kolarov and Sergio Aguero cutting the deficit to a solitary strike. It was an impressive resurgence, all the more startling given City had had Vincent Kompany sent off for a two-footed lunge at Nani in the first half. It was a contentious decision from referee Chris Foy – the Belgian defender had left the ground but had taken the ball rather than the United winger – but it served only to endorse City's belief that they can continue to discomfit United this season.

"At Old Trafford, with 10 men for 40 minutes, they lost 6-1," said Roberto Mancini, the City manager. "Today we played for 80 minutes with 10 players, scored two goals and didn't concede any chances. If we have the same attitude and same strength every game, we will win the league."

Bereft of the likes of Mario Balotelli through injury and Kolo and Yaya Toure, who are due to depart for the Africa Cup of Nations, United will be aware that their rivals across the city will only get stronger as the season progresses. It was down to their own perceived lack of depth that Ferguson had completed the re-signing of Paul Scholes ahead of the trip to the Etihad Stadium, the United midfielder having retired at the end of last season.

"The last few weeks, Paul has been training very hard with the reserves and doing a lot of work in the gym," said Ferguson. "He came to see me and said 'I regret retiring'."

He may have been yesterday rethinking that decision shortly after he made his appearance from the substitute's bench after an hour. It was Scholes' slack pass which allowed Aguero in to stab a shot past Anders Lindegaard, 15 minutes after Kolarov had given City hope with a rasping free kick which had passed the United goalkeeper before he could move. He sent another drive at Lindegaard late on which the Dane did well to turn away as City pressed for a dramatic equaliser.

"It was ridiculous," said Ferguson. "We got careless and made City better than they were. They went defensive in the second half and waited for us to make mistakes – and we made them. Fortunately we scrambled away with a victory. From a position where we should have battered them."

Yet the return of Scholes – who was watched by former United team-mate David Beckham, ensconced in the stands – also had a positive impact for the Old Trafford club. For one thing, when his name was read out over the Tannoy the pocket of United fans erupted in cheers, the return of the feted midfielder provoking a renewed hope within a support who have been galled by recent results and events at their club.

He also offers Ferguson – and perhaps more importantly the United squad – another source of experience to call upon in times of need. Given the season they have had so far there are bound to be a few more of those as the season reaches its climax.

They survived their latest test, though. Just.