Argentina’s players bounced in unison in front of the banks of compatriots in sky blue and white on the final whistle here, some in their number ripping off their shirts and swirling them round in celebration to mimic those in the stands. This team burst through a psychological barrier in overcoming a becalmed Belgium. Where the last eight has seen La Albiceleste stumble too often in recent tournaments, now they will contest a semi-final for the first time in 24 years. São Paulo should brace itself for an invasion.

This was a reminder of innate Argentine qualities, a streetwise display built on rugged teamwork, committed defence and adorned, at times, with flashes of spellbinding skill from Lionel Messi. The captain’s 91st appearance has drawn him level with Diego Maradona in joint sixth place on the list of those capped for this country, with his first-half display in Brasília the latest masterclass he has delivered at these finals. One mind-boggling blur of close-control as Axel Witsel, Marouane Fellaini, Daniel Van Buyten and Fellaini again snapped at his ankles on the edge of the Belgian penalty area will live long in the memory.

Admittedly he should have scored a fifth goal in stoppage time only for Thibaut Courtois to maintain his recent hex over the Barcelona playmaker by stifling his attempt – the goalkeeper has now gone eight games in succession against Messi without conceding to the little master – but, in the end, it mattered little. There was satisfaction to be had that Alejandro Sabella’s team, for once, had not been entirely reliant upon their captain to force their progress. Gonzalo Higuaín scored the only goal of the game. Javier Mascherano and Lucas Biglia were industrious in breaking up play in defensive midfield. Their rejigged back line, sometimes suspect, heaved and succeeded in retaining the clean sheet. Their reward was a place in the last four.

Only in the frantic latter exchanges did the Argentina rearguard hint at cracking, fatigue and cramp kicking in as Van Buyten joined the substitute Romelu Lukaku up front as battering rams with the Belgians flinging themselves forward in desperate pursuit of an equaliser. Yet the closest they came was arguably Witsel’s volley over the bar, the attempt going the same way as Fellaini’s earlier header.

Within seconds, the South Americans had their relief. “I’m so happy for the boys and for the people of Argentina that we can be back in the best four in the world,” offered Sabella, albeit that joy was not entirely borne out over the course of his rather downbeat and growled post-match assessment. One Argentinian journalist’s request that he join him in song went down particularly badly.

He was more comfortable in praise of his players. “Messi played a wonderful match,” he said. “It’s not only about scoring goals. It’s having possession, taking out three opponents, and every move he makes is a sign of hope for us and endangers our opponents. The influence he has on the pitch is decisive. He never loses the ball – or almost never – and occupies three players every time he has it, so he gives us water in the desert. Today, when the terrain was dry, he gave us that breath of fresh air every time he held on to the ball.

“But that was the ideal team performance: giving everything for the man alongside you. They don’t only have their own number on their backs, but they wear the numbers of every player in the team. They’ve made history in a way, going down among the best four in the world. And we’ll see if we go on and achieve more. We’ve achieved our minimum objective, to reach the semi-finals. Now we want more.”

It required only one early moment of penetration to deflate Belgian optimism. Vincent Kompany strode upfield only for a heavy touch to surrender possession – the centre-back’s mistake noteworthy for being so rare – to Mascherano, who shifted the ball on to Messi. The forward gathered, pirouetted and dizzied Kevin De Bruyne and Fellaini before feeding Ángel di María, whose pass into the area might have been innocuous had it not deflected off Jan Vertonghen. The flick off the full-back wrongfooted Belgium’s defence, but Higuaín’s sweeping first-time shot from the edge of the area across Courtois and into the corner was still magnificently executed. The Napoli striker had laboured through this tournament up to that point, his confidence shattered in front of goal, but instinct took over when it counted. He dedicated the winner to Alfredo di Stéfano, who is in critical condition back in Madrid after reportedly suffering a heart attack.

The loss of Di María to a thigh injury provided Argentina’s reality check – the player tore the muscle as he attempted a first-half shot – but they were able to disrupt with their advantage established. They actually created the better chances, Higuaín nutmegging Kompany after the interval and then clipping the bar with his attempt before Messi’s opportunity in stoppage time. Belgium, so impressive on paper and in periods in reaching their second ever quarter-final, felt diminished by the occasion, with only Vertonghen’s delivery from the left offering threat. Eden Hazard was peripheral throughout and was substituted before the end. He, like this team, will hope to have a more lasting impact at Euro 2016.

They were crestfallen at the end, Van Buyten departing the arena close to tears with this surely the 36-year-old’s last involvement at a major finals. Marc Wilmots’ complaints about his opponents’ style and tactics rather ignored the reality that his own charges had spluttered when an opportunity had been there for the taking, yet their biggest threat at the Estádio Nacional remained their potential. Their time may still come. Argentina’s could yet arrive in Brazil in the eight days that remain of this tournament.