How often can a side flirt with elimination before they finally go out? Ivory Coast had played the entirety of the second half with 10 men, were 1-0 down and seemingly on their way out of the Africa Cup of Nations when, in the final minute of normal time, the substitute Simon Adingra darted in from the right flank, squared for Seko Fofana and then, after inadvertently blocking his teammate’s shot, reacted quickest to stab in the loose ball. Another great escape for the hosts, who completed their victory with characteristic drama in the final minute of extra time.

It may not make much rational sense, but Ivory Coast are in the semi-final, where they will face Democratic Republic of the Congo. If any side can stop them, they will have to overcome not only a fine squad and a passionate home support, but also a sense of narrative momentum that, at the moment, feels unstoppable. Mali were devastated and while some of the players and their coach, Éric Chelle, wept, others jostled the Egyptian referee, Mohamed Adel, leading to a red card for Hamari Traoré.

Related: Mali 1-2 Ivory Coast: Hosts win AFCON classic at the end – live

Ivory Coast’s problems in making it through the group not only added to a sense that there is something otherworldly about this side – les Revenants, as locals have taken to calling them – but also meant the hosts having to move away from the Stade Alassane Ouattara in Abidjan. After a last-16 tie in Yamoussoukro, there was profound resonance in the venue for the quarter-final: Bouaké, the capital of the rebels during the first Ivorian civil war.

The South Africa goalkeeper Ronwen Williams saved four penalties in a shootout against Cape Verde to send his country into the last four of the Africa Cup of Nations after a 0-0 draw in their quarter-final on Saturday.

Mothobi Mvala scored the decisive kick to make it 2-1 on penalties and seal South Africa's first semi-final place since 2000 but they were fortunate to get past the smallest nation in the tournament at Charles Konan Banny Stadium. They will face Nigeria in Bouake in their semi-final on Wednesday.

In a game full of missed opportunities Cape Verde, who have a population of 600,000 but draw most of their players from a large diaspora in Europe, had most of the chances and were only denied a last-four place by Williams, who made a brilliant save to take the game into extra time before continuing his heroics in the shootout.

South Africa dominated the early exchanges but went off the boil after the cooling break midway through the half, and it was Cape Verde who finished the first half strongly.

They could have been ahead had their captain, Ryan Mendes, taken his chance as the ball fell favourably for him in the box but after several touches to get into a shooting position, his effort was blocked.

There was an even better opportunity for Garry Rodrigues when he was free on goal after breaking down the left but he hit a soft shot straight at Williams.

Another chance for Rodrigues in the 70th minute was blocked, while Kenny Rocha and Jovane Cabral also came close minutes after.

But they should have sewn it up in stoppage time when Gilson Benchimol got goalside of the defenders and hit the ball perfectly on the half volley only for Williams to make an outstanding save, deflecting the ball on to the woodwork and collecting the rebound.

Cape Verde’s veteran goalkeeper Vozinha had little to do in the 90 minutes but made two excellent stops at the start of extra time, first a point-blank save from Mihlali Mayambela and then turning a header from Evidence Makgopa over the bar.

Vozinha also made one save in the shootout but it was Williams who emerged as the hero.

In 2007, Didier Drogba took his African player of the year award to Bouaké as a gesture of reconciliation before persuading the government and his teammates to play a Cup of Nations qualifier against Madagascar there, which is why this is called the Stade de la Paix. Ouattara, who became president in 2011 after the second civil war, had spoken of the tournament as a symbol of the unity of the modern Ivory Coast, but that is an easier story to tell when the team is playing in the Islamic north as well as the Christian south.

Political symbolism aside, the story of Ivory Coast in this tournament has been of a team leading a charmed life. The assumption had been they were out after they followed their defeat by Nigeria with a 4-0 humiliation against Equatorial Guinea, but then Ghana conceded twice in added time against Mozambique and Zambia failed to score against Morocco to let them through as a best third-place team. They were 1-0 down to the defending champions, Senegal, with four minutes remaining in the last 16 before pinching an equaliser from a penalty and winning the shootout.

Here they seemed overwhelmed early on by the physicality of Mali, nobody more than the Bayer Leverkusen defender Odilon Kossounou. Having got away with one handball in the box because of a VAR offside, he did then give away a penalty, hauling down Lassine Sinayoko. Yahia Fofana, though, made a fine save low to his left to deny Adama Traoré. Kossounou then collected a second yellow card just before the break after another foul on Sinayoko.

Emerse Faé, in only his second game as a coach having been appointed when Jean-Louis Gasset was sacked at the end of the group stage, settled into a deep-lying 5-3-1 that restricted Mali to long-range efforts but one of those, a 25-yard strike from Nene Dorgeles, flashed into the top corner. Given the 21-year-old was born in Mali to Ivorian parents, and refused to celebrate, it seemed the narrative drive had met a formidable narrative block. But les Revenants came back again; even with 10 men, even a goal down with seconds remaining, they simply will not die.

Sébastien Haller hit the bar but the match was won in the final minute of extra time as Oumar Diakité deflected in a Fofana drive with a deft flick of the heel, before himself being sent off for his celebration. Faé’s implausible fairy story goes on.

In the evening’s other quarter-final, South Africa beat Cape Verde 2-1 in a penalty shootout after playing out a goalless draw over 120 minutes. The South Africa goalkeeper Ronwen Williams saved four spot-kicks to help his side book a place in the semi-finals, where they will face Nigeria.