Lionel Messi’s Argentina side kept alive their bid to become the ‌first team to retain the World Cup since Brazil in 1962 with a victory that sparked pandemonium among the sea of sky-blue-and-white supporters and ended Switzerland’s fairytale run.

The Swiss had reached the World Cup quarter-finals for the first time since 1954, but fell one match short of an unprecedented semi-final appearance, suffering a major blow when Breel Embolo was sent off for diving in the 72nd minute.

The thriller — Kansas City’s final World Cup game — looked headed ​for a shootout before Alvarez, excellent all night, whipped a gorgeous long-range shot into the top corner in the 112th minute that Gregor Kobel at full stretch had no ​chance of saving.

Martinez added a third for the holders nine minutes later when he calmly slotted home the rebound of Thiago Almada’s shot.

“We’re used to ⁠suffering,” Almada said.

“We come from being champions and now being among the top four is not easy at all. We want to chase glory and put Argentina at the very top.”

Argentina take early lead

Alexis Mac Allister had given Argentina the lead in the 10th minute, meeting Messi’s corner at the front post with a flicked header that sailed beyond Kobel and into the ​far corner.

It was the first time Switzerland had trailed in the tournament.

But the Swiss — without injured leading scorer Johan Manzambi, who had three goals and two assists — refused to wilt.

After a largely uneventful first half, the contest burst into life as Switzerland repeatedly tested Emiliano Martinez, forcing the Argentina goalkeeper into a string of sharp saves and jolting the South American supporters into full voice.

An equaliser felt almost inevitable as the Swiss piled on ​the pressure, and they were rewarded in the 67th minute when Dan Ndoye exchanged passes with Ricardo Rodriguez down the left before sliding a right-footed finish through Martinez’s legs.

Ndoye snarled into ​the nearest television camera as the small section of Swiss fans erupted.

However, Switzerland were reduced to 10 men when Embolo got a second yellow card for simulation.

It came after a lengthy VAR review for mistaken ‌identity, with ⁠the referee overturning his initial decision to caution Argentina’s Leandro Paredes.

The call left Swiss coach Murat Yakin livid.

“I know that (FIFA) will protect their referee, but this rule destroyed our game today, and it is incredibly painful,” he said.

“And to be eliminated in that way hurts a lot. Unfortunately, we have to accept it.”

Embolo collapsed in tears and was consoled by teammates as he left the pitch.

“This team has so much passion. Despite the fact we were one man down, we gave it all. I’m extremely proud of everyone,” Swiss coach Murat Yakin said.

Switzerland pinned back

Argentina responded to ​Swiss misfortune by pinning them back for long ​periods.

Cheered on by fans who made ⁠Arrowhead Stadium feel more like Buenos Aires than Kansas City, they surged forward as chants of “Vamos, vamos!” echoed around the venue.

Messi, playing in his sixth World Cup at the age of 39, nearly settled the contest in stoppage time with a fierce strike that flashed inches wide ​of the post.

Switzerland weathered wave after wave of pressure and appeared destined to force a shootout, but their resistance was finally broken in ​the closing moments of ⁠extra time.

“We had to suffer a lot. We knew they were a physical team and that gave us a lot of trouble,” said Argentina boss Lionel Scaloni.

“Today we had luck on our side because one of their players was sent off. We could have played better, but it’s a big achievement to be in the semi-finals.”

The distraught Swiss dropped to their knees, consoling each other, while Argentina’s star-studded ⁠squad lingered ​on the pitch soaking up the atmosphere as Messi tugged off his shirt and twirled it above his head ​in front of the Argentina fans.

Wednesday’s semi-final will mark the first time England have faced Messi at a World Cup.

“It doesn’t matter whether it’s England or Norway (who lost 2-1 to Thomas Tuchel’s England after extra time earlier on Saturday),” Scaloni ​said.

“We’re going to come up against a team that plays very well and has a great coach.”