Italy have spent the last few years veering between ignominy and glory. When embarrassment beckoned, a glorious goal instead lent salvation. For the 43 minutes when they trailed, the holders confronted the possibility of an early exit, of expulsion at the first stage.

Instead, almost certainly, Croatia will be cast out of Euro 2024. The team which can excel over 120 minutes and penalties missed a penalty and went out to a 98th-minute goal. When it was shaping up as another classic night in the career of an all-time great, Luka Modric, he was instead upstaged by a newcomer to such occasions.

When Mattia Zaccagni came off the bench, the 29-year-old forward had never scored for Italy. When he did, it was magnificent, the Lazio winger curling a shot around Dominik Livakovic.

Italy’s prize for a lovely leveller is a tie with Switzerland in Berlin and a reasonable route to the quarter-finals. Croatia wait to see if they will rank among the four third-best teams, but they probably won’t.

They can rue two injury-time goals against them: by Klaus Gjasula for Albania and, still later, Zaccagni for Italy. Six points became two; the final blow was the cruellest, probably the most decisive. An extraordinary evening had an incredible end.

For Modric, a mad 33 seconds, brought extremes of emotions. Missing a penalty, scoring a goal, he seemed the match-winner when he was substituted to a standing ovation. He could only watch as, in all likelihood, his tournament ended. His goal had been redemption, a rescue act. Yet it was not enough. A wild night had a wonderful end for the Azzurri.

Italy, the Euro 2020 winners but the team who failed to qualify for the last two World Cups, continue the defence of their title; not convincingly, but euphorically. They still have not beaten Croatia, yet a status as their bogey side may need re-examining after this. It was Zlatko Dalic’s team who were left feeling heartbroken, the experts in escapology experiencing how they have made others feel with their litany of comebacks.

For Luciano Spalletti, it was a game of two substitutions: first sending on Davide Frattesi, whose arrival backfired, and then Zaccagni, the unheralded saviour. When Andrej Kramaric’s shot hit the outstretched arm of Frattesi, referee Danny Makkelie was sent to the monitor and pointed to the spot. Bruno Petkovic had missed for Croatia from 12 yards in their defeat to Spain. Donnarumma had won Italy Euro 2020 in a penalty shootout. And when Modric sidefooted his spot kick to the goalkeeper’s left, a giant dived and parried.

And yet the response was still more remarkable. Donnarumma performed more heroics by parrying Budimir’s shot but first to the rebound, lifting his effort into the roof of the net, was Modric. It was reward for Zlatko Dalic, who had brought on the penalty-box presence of Budimir at the break, after initially doubling down on a theory and starting with the passer Mario Pasalic in the front three. Yet this was the kind of goal from Modric that spoke more of his willpower than the technical talents that have propelled him to six Champions League titles. He looked more like predatory striker than stylish passer.

When he struck, Leipzig could have been Zagreb. There were flares going off, drinks thrown everywhere. It was bedlam. For Italy, this had the feel of an away game in the Balkans. Croatia’s population is famously small, less than four million, but the sense is many of them have decamped to Germany.

Initially, they saw a cohesive performance but with little to celebrate. Donnarumma tipped Luca Sucic’s long-range effort over.

Otherwise, as Spalletti’s switch to a back three brought a sense of solidity, the better chances before the break fell to Italy. They looked to use their aerial ability, sensing a Croatian frailty against crosses. Brought in for Gianluca Scamacca, Mateo Retegui headed wide. Alessandro Bastoni came closer, drawing an acrobatic save from Livakovic when found by Nicolo Barella.

Bastoni threatened an equaliser, too, but headed over. Italy had made a watchful start, with Spalletti’s new-look defence indicating they could play for a 0-0. He ended up throwing on three forwards and Scamacca almost connected with Federico Chiesa’s cross. Croatia had started to look comfortable, Italy ragged.

Then came Zaccagni’s wonder strike. And Italy, the side who lost World Cup play-offs in 2017 and 2022, had passed their test of nerve spectacularly, brilliantly, perhaps luckily. Donnarumma, the hero of Euro 2020, can point to other saves that made a difference.

Rather than Italy, it is Croatia who are left in purgatory. They have to wait for Denmark and Slovenia, Czech Republic and Turkey, hoping for favours. But Italy may have achieved what long seemed impossible in tournament football. They might have killed off the Croatians.