The Dutch are in uncharted territory. Before their Women’s World Cup quarter-final against Italy they had never been this far at Fifa’s showcase tournament. Indeed, France 2019 is only their second World Cup appearance, after they made their debut four years ago.

Yet the European champions displayed no inexperience as they navigated a robust Italian test in Valenciennes to secure a historic semi-final spot. Second-half headers from Vivianne Miedema and Stefanie van der Gragt – the first goals Italy have conceded from open play this tournament – were too much for the Italians.

“The word proud is more suitable than surprised,” said the victorious coach, Sarina Wiegman. Miedema was more forthright. “I did not expect to be in the semi-finals of this World Cup,” she admitted.

The all-European clash began in sweltering mid-afternoon conditions, as the heatwave gripping northern France refused to abate. Every other match in this round had been scheduled as an evening fixture but the Italians and the Dutch drew the short straw. The Royal Dutch Football Association even wrote to Fifa asking if the match could be delayed, to no avail. “We will feel the heat,” joked Italy’s manager, Milena Bertolini, on Friday.

They did – via the baking sun and their dangerous opposition. From the opening whistle, Italy endured relentless waves of pressure as the Netherlands marauded forward. Yet the Italian defence stood firm, impervious to Dutch raids.

Indeed it was Italy who had the first clearcut opportunity, with Milan’s Valentina Bergamaschi scuffing a golden chance after 18 minutes. Replays indicated there may have been an earlier offside, continuing a trend that has seen four Italian goals disallowed at the tournament. But Bergamaschi’s misplaced shot meant no VAR intervention was necessary.

The opportunity swung momentum and soon Italy were ascendant. A water break on the half-hour gave the Dutch a short reprieve, although there was no rest for either set of travelling fans – a small Italian contingent remained in full-voice as they faced down a stadium full of orange-clad Dutch supporters. Fans of the Oranje have been a common sight throughout the World Cup but they were particularly numerous in Valenciennes; less than 150km from their homeland’s southern border, the industrial French city temporarily felt like a Dutch exclave.

The Netherlands began the second half as they had started the first, imposing themselves on Italy’s back four but failing to find the right key to unlock it. The Dutch slowly intensified their pressure, Daniëlle van de Donk at one stage curling a long-range effort against the bar.

Tensions flared within both teams, mainly in the direction of the Uruguayan referee, Claudia Umpiérrez, who dealt several yellow cards as the literal and figurative temperature increased. “When you are tired, you lose your mind and start to make bad decisions,” said Bertolini.

At long last, after 70 minutes, a goal arrived. Sherida Spitse lofted a free-kick to Miedema, whose glancing header past Laura Giuliani’s outstretched glove gave the Arsenal star her 61st national-team strike. Drenched with sweat, the Italians looked despondent as their miracle run to the quarter-finals at their first World Cup in two decades appeared over.

The Italian’s ill-discipline then confirmed their exit. They conceded another free-kick, this time on the corner of their own penalty area, and Spitse whipped in her second effort within 10 minutes. Van der Gragt connected with a powerful header, leaping for joy in the knowledge that she had put the Netherlands’ progress beyond doubt.

The Dutch celebrated joyously following Umpiérrez’s final whistle, in the stands and on the pitch. They now find themselves only two victories from World Cup triumph, a remarkable achievement four years after their tournament debut. The win also secures them a ticket to the Tokyo 2020 Olympics. “That is a special feeling,” said Miedema. “I have wanted to play at the Olympics since I was young.”

Crestfallen Italy received a standing ovation before Bertolini gathered her players on the pitch for a final time. “There were tears,” she said. “The dream is over.” But the Italian football hall of fame inductee remained upbeat. “I told them I was very proud of what they had achieved – they have allowed the public back home to fall in love with the game. We have done something extraordinary.”