Manchester City clambered back from two goals down to reach the FA Cup semi-finals after Sergio Agüero jumped off the bench to head home a controversial winner and break Swansea hearts.

Agüero was marginally offside when he snatched victory two minutes from time, stooping to nod in Bernardo Silva’s cross, but in the absence of the video assistant referee Pep Guardiola’s side completed a stirring comeback to keep hopes of an unprecedented quadruple alive.

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The Premier League champions’ equaliser was equally contentious, with a soft penalty awarded for an alleged foul on Raheem Sterling. Agüero’s spot-kick smacked a post and went in off the backside of the excellent Kristoffer Nordfeldt, but the manner of victory left Guardiola feeling uneasy and Swansea with a sour taste in their mouths.

“Why is VAR not here? It’s a question for the authorities,” said an apologetic Guardiola, in reference to the FA’s decision to utilise VAR only at Premier League stadiums in the Cup this season citing consistency, despite the technology having previously been used here. “If it is not a penalty, and it is offside like the people say, I’m sorry. They ask all the clubs about the VAR and all around Europe the VAR is there and here it is not.

“I don’t like to win games when the decisions are wrong but the referees don’t want to make the wrong decisions – they must be helped. It’s centimetres. Sometimes the VAR takes three minutes to make a decision so imagine taking that decision in one second during the game. It is not easy.

“But at the same time we played incredibly well, with the amount of actions, the amount of crosses and the [Swansea] goalkeeper was incredible. We will analyse the game – we deserved to win by far – but to win in that way, I am sorry.”

Up until the 69th minute, when Bernardo Silva pulled a goal back with a sweet first-time strike, the chasm between the sides was not apparent. For Guardiola there was a sense of deja vu, given his side surrendered their aspirations to land a quadruple in the fifth round at Wigan last season.

A beautiful, sweeping, one-touch Swansea move culminated in Bersant Celina, who spent six years on the books of Manchester City, curling a shot home to open up a two-goal buffer. Swansea were daring to dream, just as their manager, Graham Potter, had asked them to do. “Can we play you every week?” was the joyous verse from home supporters, until Guardiola asked his A-team to step off the bench and dig City out of a hole, summoning Agüero and Sterling around the hour mark. Eventually Swansea buckled but, as their manager said afterwards, this was a courageous performance to be proud of.

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Guardiola’s side have already won on three occasions at Wembley this season – lifting the Community Shield and Carabao Cup either side of a league win over Tottenham – and will head there for a fourth time next month. A congested April schedule comprising at least seven matches, including a two-legged Champions League quarter-final against Spurs, will be season-defining. “When you fight for all the titles, you don’t have rest and every three days there is a game. It is so, so healthy to win these kind of games,” Guardiola said. “You need these kinds of situations to go through, because there are a lot of games and different competitions every three days. Our physicality and mental approach is so demanding. We know now every game we lose, we lose the competition.”

Swansea had been flustered by City’s early intent but then Fabian Delph, too, hit the panic button, fouling Connor Roberts in the box. Celina, who badly fluffed his lines from 12 yards at West Brom on Wednesday when his attempted panenka ended up a viral misery, passed on penalty duties to Matt Grimes, who made no mistake. Then came a spot of Celina magic, an arcing first-time finish beyond Ederson to complete a slick move. It had been coming too, with Daniel James’s electrifying pace causing havoc; the winger humiliated Nicolás Otamendi in one breath and left Kyle Walker – no slouch himself – for dead in another. All the while, Guardiola’s hands were rooted in his pockets.

As Guardiola hopped down the touchline, his players gradually upped the ante. The arrival of Sterling and Agüero heightened the anxiety rippling through the Swansea defence and when Roberts failed to clear Walker’s cross, Bernardo Silva found the top corner. Sterling should have levelled but spooned over before falling under pressure from Cameron Carter-Vickers, with the referee Andre Marriner pointing to the spot. Agüero’s penalty was hardly perfect but City had completed the comeback. And then came the winner. Moments after a sublime double save by Nordfeldt to deny Agüero, the Argentinian nodded in at the near post.

“Manchester City almost play a different sport to what we play,” Potter said. “They are relentless with their quality.”