Four years ago, it was the nightmare in Nice. This time it was the celebration in Sydney. Where lapses of mental fortitude against Norway at the 2019 World Cup led to the Matildas exiting the tournament in the last 16, maturity and toughness saw them through against Denmark on Monday.

Without one, the Matildas may not have managed the other. This was a journey that began in the bowels of the Stade de Nice on a French summer’s night with the captain, Sam Kerr, in tears having missed their opening penalty of the shootout. In that moment, the striker made a commitment to her country: “We’ll come back.” She was true to her word.

Four years later, Kerr found herself in the concrete alleyways of another gargantuan stadium, this time at the end of a wintery Sydney night. Where Nice ended in tears, Sydney offered jubilation. The Matildas banished those World Cup demons to book a slot in the quarter-finals for the fourth time. “We were talking about Nice today,” Kerr said.

She paused for a moment, reflecting. “There was no way we were going to penalties tonight,” she said with a laugh. “Because we wanted to get the job done.”

That feels like an accurate summary. It was not always pretty. It was certainly less flashy than the devastating 4-0 win against Canada, the Olympic champions, a week ago. But that only made the victory even more impressive.

Denmark were no easy opposition – they took the game to the Matildas early and made life difficult. But Australia still found a way to win. That is a quality possessed by teams that go deep in tournament football. The ability to win when it counts.

The Matildas have always been able to turn up when they are underdogs, when they have their backs to the wall – just as the team did in the must-win final group stage match against Canada. They are not the “never say die” Matildas for nothing. But historically they have been less capable of game management when anticipation is high and victory is expected.

It felt, in the first 18 minutes, that this may be another encounter in the latter category. The Matildas entered as favourites but quickly found themselves pinned back, Denmark’s captain, Pernille Harder, floating around the attacking third with a lethal edge. The opportunities kept falling Denmark’s way; nerves rose in the capacity crowd as a goal beckoned. Any one of those moments could have killed the Matildas’ World Cup dreams.

Then the wing-back Steph Catley went down and stayed down. While she received medical attention, the rest of the team rushed to the sidelines for a briefing from their head coach, Tony Gustavsson. It felt like a timeout – and it worked. As play resumed the Matildas settled into a more comfortable rhythm, Denmark’s chances dried up and not much later Caitlin Foord scored the goal that put Australia one step closer to the last eight.

The fortuitous nature of the injury break was put to Catley. “It’s true,” she said. “Just a great coincidence.” The press, and some Matildas walking past, burst into laughter. “I’m a comedian,” Catley said.

Once the merriment subsided, she acknowledged that the delay, and Gustavsson’s intervention, had come at the right time. “Sometimes it’s a good reset when there’s a break in play,” she said.

It may seem odd, in a game of 90 minutes which featured two superb goals from the Matildas, to highlight a temporary knock. But the defender’s decision to take her time, to receive medical treatment, to allow her teammates to regroup, showed wisdom – gameswomanship even – this team have not always possessed. At the end of a long but successful night, “mature” was a common adjective for the performance. In Nice, the moment got to the Matildas; in Sydney, the side was a model of composure.

So the Matildas march on – to Brisbane for a meeting with France or Morocco. There was another commonly repeated phrase on Monday. “We’re taking it one game at a time,” Kerr said. It has become this team’s mantra. And fair enough. “We’re not going to look too far ahead because that’s when you slip,” the striker added.

But when a team keep taking things one game at a time, sooner or later they end up deep in the tournament. The Matildas have never gone beyond the quarter-finals of a World Cup. One more win takes them into history. Another secures a berth in the final. Three more matches stand between the Matildas and a triumph that would reverberate around this increasingly football-mad country and across the globe.

Gustavsson is a maths teacher by training and loves nothing more than reeling off statistics and tactics – plenty of that would come. But he was asked something more philosophical, about how the team had changed as humans. The question was directed to the adversity of the past three weeks, but his answer aptly summed up the Matildas’ journey from Nice to Sydney.

“We all as human beings grow every single day,” he said. “We learn and we change and hopefully improve. We learn from experiences.”

The Matildas learned from the experiences of four years ago. That is how they vanquished Denmark. That is why they are increasingly the team to beat.