Harry Kane celebrates with his England teammates in front of the jubilant fans at Wembley after the 2-1 victory.

Sometimes, your luck is just in, you catch a break when you most need it and, after so much major tournament semi-final heartache, England finally got something to go their way and, in the process, one of these suffocating high-tension encounters to follow suit.

The 90 minutes had been nerve-shredding, England forced to find a response to Mikkel Damsgaard’s stunning 25 yard free-kick on the half-hour, which they did when Bukayo Saka’s cross, intended for Raheem Sterling, was bundled over Denmark’s goal line by their captain, Simon Kjær.

With the battle lines drawn, England pushed in the second half of regulation time, but could not create too much of clear-cut note.

Denmark’s resistance was stout. But on into extra time, it finally happened – England had the chance to win only their second semi-final in six attempts and advance into a first final since the 1966 World Cup.

Inevitably, it was Sterling who sparked it, having worried Denmark with his pace from the first whistle. He sliced into the area and went to ground under a challenge from Joakim Mæhle.

The contact appeared minimal – to say the least – but the Dutch referee, Danny Makkelie, felt it was sufficient to award the penalty and VAR agreed. To compound the uneasiness, there was a second ball on the pitch at the time, although it did not appear to have affected play.

So there was Harry Kane, standing over the kick and, as everybody knows, he hardly ever misses. The England captain has a reputation for ice-cool efficiency for good reason. And yet he did miss this one, his side-footed shot read too easily by Kasper Schmeichel, who got a hand down to it.

But the Denmark goalkeeper, who had earlier made saves to deny Sterling and Harry Maguire, did not get the ball away. He merely got it out and, eyes lighting up in relief, Kane gobbled up the rebound, lashing into the empty net.

Harry Kane scores from the rebound for the winner in extra time after seeing his penalty saved by Kasper Schmeichel.

Harry Kane scores from the rebound for the winner in extra time after seeing his penalty saved by Kasper Schmeichel. Photograph: Laurence Griffiths/Getty Images

How Wembley exploded. The biggest crowd at a British sporting event in 16 months had provided the noise throughout, living every moment and now they knew. The night would belong to them, to Gareth Southgate and a squad that has ripped up the old narratives around the England team. At last, Denmark were broken. They would finish with 10 men after the substitute Mathias Jensen was injured in the 105th minute with all of the replacements having been used. They had nothing left.

Southgate’s game management has hardened since the misery of allowing a 1-0 lead to become a 2-1 extra-time defeat by Croatia in the 2018 World Cup semi-final. Here, he took off Jack Grealish, who had only entered as a 69th minute substitute, and sent on Kieran Trippier, asking him to play at right wing-back and switching from 4-3-3 to 3-4-3.

England had suffered. They always do. But in the second period of extra time there was an unusual sense of comfort as Southgate’s players closed out the game, keeping their composure, taking care of the ball. The statistics showed that Denmark touched it only once inside the England area in those final 15 minutes.

And so at full time, there was the sight of Southgate advancing towards one of the goals and, with his fists clenched, yelling in delight at the fans behind it. Was this closure for him? He has lived with the pain of missing that penalty in the shootout defeat against Germany here in 1996 at the same stage of this competition. Now fortune had favoured him and his team from the spot.

England deserved it. They had considerably more shots than their opponents and, from an early point in the second half, it was they who made the game.

Southgate and his players continue to make pieces of history, to tear down psychological barriers. Not since the 1960s have England reached a final and a semi-final in back-to-back tournaments while, in previous European Championships, they had only ever won one knockout tie – against Spain on penalties in 1996. They will back themselves in Sunday’s final against Italy because there is a hard-won steeliness to their mentality.

Denmark captain Simon Kjær inadvertently steers an equaliser for England into his own net under pressure from Raheem Sterling.

Denmark captain Simon Kjær inadvertently steers an equaliser for England into his own net under pressure from Raheem Sterling. Photograph: Alex Morton/Uefa/Getty Images

The trigger for the victory came in adversity, as Jordan Pickford was forced to retrieve the ball from his net for the first time at these finals. England had started on the front foot, with Sterling’s speed and directness a threat, and it was he who had England’s best early chance, cutting inside after a Kane pass only to scuff the shot.

Back came Denmark. Pierre-Emile Højbjerg and Kasper Dolberg went close – the latter after a poor Pickford clearance, which would not be the only one from him – and England were in trouble when Damsgaard summoned whip and power on his free-kick. The ball fizzed past the flailing Pickford.

Southgate had promised that, if and when England found themselves behind, they would not enter panic mode as they had done in the notorious Euro 2016 defeat by Iceland. So it proved.