The idea for Manchester City had been to channel the anguish from the Champions League quarter-final exit against Real Madrid into something more positive and it surely applied most strongly to one of their number.

Bernardo Silva was a snapshot of negative emotions after his terrible penalty in the shootout with Madrid on Wednesday, the chip that went so horribly wrong turning the tide against his team. This is what redemption looks like.

Related: Manchester City v Chelsea: FA Cup semi-final – live

It was a curious performance from City as they fought to keep alive their hopes of a silver-lined finish to yet another season. The double-treble might have gone but the double- Double had not. Chelsea were threatening, they created gilt-edged chances – mainly for Nicolas Jackson – and they were the better team in the second half.

What was most striking was City’s lack of control. They were loose, jaded – a legacy of the 120-plus minutes against Madrid and the penalties, Pep Guardiola making his feelings perfectly clear afterwards about what he called the unacceptably quick turnaround for this showpiece FA Cup semi-final.

Guardiola said he did not know how City survived; in terms of overcoming adversity, it was – in his opinion – one of his champion team’s finest days. But City survived because, unlike Chelsea, they know how to make their punches count. There were 84 minutes on the clock when they landed the decisive one.

Kevin De Bruyne had been a symbol of their performance; trying everything and not seeing it all come off, by any means. He was not the only City player to lose easy balls. He was, however, a repeat offender.

De Bruyne kept going and it was his cutback from a pass up the inside left from the substitute Jérémy Doku that teed up Silva. The ball flicked off the leg of the Chelsea goalkeeper, Djordje Petrovic, and there was Silva, opening up his body to guide home with the help of a deflection off Marc Cucurella.

It was a lesson in composure where it matters the most and one that Mauricio Pochettino is desperate for his Chelsea players to learn. His team have regularly been a match for the Premier League’s top clubs, drawing twice against City and once with Arsenal and Liverpool. So they were again here, showing that they have the quality. A lack of ruthlessness is what has held them back.

When one substitute, Ben Chilwell, delayed for too long towards the end while another, Raheem Sterling, screamed for the cross in front of goal – along with every Chelsea fan – it was of a piece with the defining moment of the first half.

It was in the 29th minute, Enzo Fernández had sent Jackson clean through and it looked good for Chelsea when the striker dropped his shoulder and went around Stefan Ortega. But then he hesitated and did not shoot. He checked inside and the chance was lost. The pass for a teammate was never on. Fast forward to the Chilwell moment. When he finally released the ball, it was easily cut out.

Chelsea had got away with a few near-misses at the outset when trying to play out from the back and into the City press. But they grew into the game, Cole Palmer shimmering with menace in the No 10 role, Conor Gallagher bringing the hustle out of position of the left of midfield.

Jackson was excellent until the final action, catching the eye with his sharp turns and electric pace, and Chelsea got forward with some slick one-touch stuff in the first half. Noni Madueke had a shot blocked by John Stones and Palmer almost worked his magic in the 37th minute. He showed his silky skills and movement to glide past Rodri and open up the shooting opportunity. Ortega got down and across to push it away.

Guardiola, who was without the injured Erling Haaland, had started Phil Foden as one of his central midfielders. It was he who might have given City the lead in the 15th minute after he ran on to a De Bruyne pass. Foden had only Petrovic to beat but his first touch was slightly heavy, making the angle too tight and the chance was gone.

Julián Álvarez almost created something with a piece of quick-footed trickery on 21 minutes while Silva was denied by a Cucurella clearance, although Foden was flagged offside in the middle. Chelsea could take heart from the statistic that showed City had zero shots on target before the interval.

It was not difficult to imagine what Pochettino was thinking at the start of the second half when Gallagher released Jackson but he could not beat Ortega, the keeper making a big block. On the second phase, Palmer crossed from the right and Jackson headed at Ortega from point-blank range.

• Download the Guardian app from the iOS App Store on iPhone or the Google Play store on Android by searching for 'The Guardian'.

• If you already have the Guardian app, make sure you’re on the most recent version.

• In the Guardian app, tap the Menu button at the bottom right, then go to Settings (the gear icon), then Notifications.

• Turn on sport notifications.

It felt as though it was on for Chelsea, who appealed loudly for a penalty when Palmer smacked a free-kick into the City wall, the ball seeming to strike Jack Grealish’s hand. Then there was the moment when Palmer turned Manuel Akanji and released Moisés Caicedo. No prizes for guessing what the end product was like.

At the other end, Foden and Doku worked Petrovic while De Bruyne steered off target. It was easy to think that penalties could be required. Silva was in no mood for that.