The statistical history of Belgium's Euro 2016 makes a beautiful diagrammatical sandwich – a thick middle section of effortless, almost arrogant superiority, neatly bookended by total disaster. They started by conceding two goals against Italy. Then they scored nine unanswered goals in a row against Ireland, Sweden, Hungary, and Wales. Then they let in three goals to Wales, and they went home.

Marc Wilmots had one of the most talented squads in the competition and it so happened that defeat to Italy in the first game landed his team in the easier half of the draw. He had taken the opportunity of the win against Ireland to round on critics in the media and mock them for doubting his team's togetherness. But as those Belgian reporters had suspected, Wilmots had all the parts but no idea how to put them together. Once Wales started to throw punches in Lille, Belgium's glass jaw shattered into a million pieces.

For the first few minutes it looked like it would be the opposite story, with Wales being bundled out of the competition by the first A-listers they had come up against. Belgium had already threatened the Welsh goal on several occasions by the time Radja Nainggolan was allowed the time and space to smash the ball past Wayne Hennessey from 30 yards.

The standout feature of those first 15 minutes was the unbelievable precision of Kevin De Bruyne, whose first time passes slid across the wet turf, beautifully weighted and geometrically perfect, as though calculated in the supercomputer brain of some kind of football Terminator. It looked as though Belgium were going to win easily.

Wales, however, had other ideas.

If you want to come back from a goal down against a better side it helps to have a couple of things going for you. First, you have to keep your heads. Second, it helps to be quick.

No team spearheaded by the still-unemployed Hal Robson-Kanu is going to be reckoned an irresistible force, but Wales have technique and speed and a terrific sense of collective purpose, and Belgium only had two of those qualities.

First Wales fought furiously to contain Belgium's momentum, an effort that cost them three yellow cards for fouls deemed cynical by the referee; then they started to play themselves into the game thanks to the calm, precise passing of Joe Allen, Aaron Ramsey and Joe Ledley in midfield.

Wilmots could sense the building danger. When Wales won the 30th-minute corner, he was at the edge of his technical area screaming at his defenders. What was he saying? As Aaron Ramsey's corner dropped over Jordan Lukaku to the unmarked Ashley Williams for a header into the corner of Belgium's net, Wilmots's meaning became clear.

De Bruyne had been on the post but had drifted inside and been wrong-footed by a header he should have cleared easily. De Bruyne took out his frustration by screaming at Lukaku, but both players were culpable.

Williams’s wild-eyed celebration showed his joy and when the match restarted it was Wales in the ascendancy. Now we saw the psychological brittleness of this Belgian team as they stood off, watching Wales pass the ball, nobody trying to win it back, forced back as though hypnotised.

When Williams got his head on another corner late in the half Wilmots's mind was made up. Marouane Fellaini was out on the field at the break warming up in his playing kit, it was obvious that Wilmots wanted his aerial power on the pitch. The player he withdrew was the lightning Atletico winger Yannick Carrasco – a hugely negative substitution given the talent available on the bench.

Nevertheless, the second half started better for Belgium, with De Bruyne, Hazard and Romelu Lukaku all going close. But once again, Ramsey proved the creator for Wales, running onto Bale’s pass and crossing for Robson-Kanu, who found himself boxed in by three Belgian defenders, but with Neil Taylor arriving in support on the left of the box.

Everyone in the stadium thought he was about to lay it into Taylor's path. Instead he produced an instant Cruyff turn that sent all three Belgian defenders the wrong way, and smashed it past Thibaut Courtois from point-blank range.

Once again De Bruyne was screaming at team-mates. He might be a football superbrain but to be a great player he’ll need to learn how to inspire his team-mates, rather than blame them. Belgium were doing a lot of screaming, mainly at the officials for imaginary fouls. The Belgians were looking for people to blame. Jordan Lukaku produced an abysmal cross from the left and was withdrawn a couple of minutes later for Dries Mertens, and Wilmots then replaced Romelu Lukaku with Michy Batshuayi as he chased the game.

Instead it was Wales' substitute, Sam Vokes, who headed in a cross from the right to make it 3-1. The Belgian fans thought this would be their year, but it's Wales who are in the semi-final of Euro 2016. Their joyful, tear-streaked supporters won't be going back to work for a while yet.