Belief. It was the overarching theme of this World Cup for the USA, from the moment they were pitted together with Germany, Portugal and Ghana in this tournament’s supposed group of death. Could they match those nations for star power and individual talent? Surely not. But that did not stop tens of thousands of Americans from making the journey down to Brazil to sing in unison: “I believe that we will win.”

They sang it in Natal, even after André Ayew’s 82nd-minute equaliser for Ghana. They sang it in Manaus, within seconds of going a goal down to Portugal. Both times the players responded, salvaging the results they required. And so the fans sang it again night in Salvador, after Kevin de Bruyne had given Belgium the lead, at what felt like the hundredth time of asking, in extra time.

But football is not a fairytale. There was no magic left in those words. Or perhaps it would be fairer to say: not enough. After Romelu Lukaku extended Belgium’s lead in the 103rd minute, there was still time enough left for Julian Green to pull one goal back – setting up a grandstand finish. The USA almost found a way, Clint Dempsey sprinting clean through at the end of an expertly worked set-piece but his close-range shot was smothered.

Credit was due to Belgium’s Thibaut Courtois for reacting so swiftly to the threat but even in victory he was not the goalkeeper more deserving of the plaudits. The USA’s Tim Howard conceded twice in Salvador but only after 93 minutes’ worth of heroic resistance. By the time this game reached extra time he had already made nine saves. At the end that figure was 15.

Americans had always known that they would need their goalkeeper to be at his best if they were to go far at this tournament. Howard has risen to the challenge. His stop from Éder on the stroke of half-time against Portugal during the group stage might just be the save of this tournament. Without it his country might not have survived this far.

He won his 103rd international cap against Germany, the most ever by a goalkeeper for the USA. He already held the record for most wins. Some would already place him alongside Tony Meola, Kasey Keller and Brad Friedel in the pantheon of American goalkeeping greats. But he came agonisingly close here to ending the debate.

His performance was on a par with Friedel’s miraculous performance against South Korea in the 2002 group stages – when the then Blackburn Rovers keeper kept out a penalty, plus three more attempts from inside 10 yards. That, to date, was the best-ever performance by an American goalkeeper at a World Cup.

Howard’s performance was every bit as special but the memory of it will always be tinged by sadness at the final result. Friedel went on in that tournament to help his team to a quarter-final, winning their first-ever World Cup knock-out game along the way. They could not repeat that achievement here.

That is a testament to Belgium, who dominated this game thoroughly. Marc Wilmots, the manager, said beforehand that he was waiting for Eden Hazard to dominate a game here in Brazil, living up to his European club form. Instead it was De Bruyne who ran the show, surging forward with persistent menace. If his own finishing had been sharper, extra-time would not have been required.

But the USA deserve credit for their approach. Even Klinsmann had understood the power of belief by the end. On conclusion of the group stage he told his players’ families to re-book their return flights for after the final on 13 July.

It was always an optimistic line but that is in keeping with the way that his team have attacked this World Cup. Despite being outmanoeuvred at times by technically superior teams, they have always tried to carry the game to their opponents when they can. Even before De Bruyne’s goal they streamed forward in numbers on the rare occasions that they could.

That approach won many admirers back home. The sight of tens of thousands of fans packing into Soldier Field – home to the NFL’s Chicago Bears – to watch this game on a giant screen will give hope to those who yearn to see the USA embrace another kind of football.

They believe that a culture shift is occurring. They believe that this is a beginning, not an end.