On the last day of group-stage action Spain benefited from two own goals as they swept through to the next round as runners-up in Group E behind Sweden, while Ukraine progressed without playing.
Having already qualified, Sweden secured top spot in their group only after scoring late to win 3-2 against Poland, for whom a Robert Lewandowski double was not enough in Saint Petersburg.
Spain had drawn their first two matches in the competition despite having home advantage in Seville, but coach Luis Enrique warned on the eve of facing Slovakia that his team were “like a bottle of cava that is about to be uncorked”.
For the second consecutive game they contrived to miss a penalty as Alvaro Morata’s effort following a foul on Koke was saved by Martin Dubravka.
However, a calamitous error by Dubravka allowed Spain to go ahead on the half-hour as the goalkeeper saw a Pablo Sarabia shot hit the bar and then palmed it into his own net as it dropped back down.
In searing Andalusian heat Aymeric Laporte then scored his first goal for Spain since switching international allegiance from France, making it 2-0 right on half-time.
The 2008 and 2012 European champions ran away with the game in the second half as Sarabia swept in a Jordi Alba cross and Ferran Torres scored within moments of coming off the bench.
Juraj Kucka’s own goal made it 5-0 as Spain equalled the biggest winning margin in the history of the European Championship.
“We’ve shut a lot of mouths up. We’ve played some great football and had more luck than in the first two games,” Laporte told television channel Telecinco.
Sweden led Poland 2-0 in Saint Petersburg, Emil Forsberg scoring an early opener and then firing in for his brace after great work from Dejan Kulusevski.
In between Lewandowski had contrived to miss a sitter, twice heading off the bar in quick succession when it seemed easier to score.
But the Bayern Munich superstar showed his worth with a stunning strike to reduce the deficit and then an 84th-minute equaliser, a scoreline that would have allowed Spain to win the group.
Poland needed to win themselves to go through but instead Sweden won it in the fourth minute of added time as substitute Viktor Claesson made it 3-2.
Meanwhile those results allowed Ukraine — who played their final Group C game on Monday — to be confirmed as through to the last 16 as one of the best third-placed teams.
Later the action concludes in Group F, as Euro 2016 winners Portugal take on world champions France in Budapest needing a draw to be certain of making the last 16.
Germany, who play Hungary in Munich, also need a draw to go through but defeat would send the Hungarians through and possibly condemn Joachim Loew’s side to another group-stage exit, just like at the 2018 World Cup.
That is unthinkable for Germany, who beat Cristiano Ronaldo’s Portugal 4-2 at the weekend to kickstart their campaign.
The build-up to the match has been overshadowed by the fallout from UEFA’s decision to block plans by Munich authorities to light the stadium in rainbow colours.
The German city wanted to protest at a law passed by Hungary’s right-wing government banning the “promotion” of homosexuality to minors.
UEFA on Wednesday added the rainbow colours to its logo while defending its decision, saying Munich’s request was politically-motivated.
European football’s governing body said in a statement that Munich’s request was “linked to the Hungarian football team’s presence in the stadium for this evening’s match with Germany.”
Munich’s mayor Dieter Reiter on Tuesday called the decision “shameful” and announced plans to decorate other city landmarks in rainbow colours instead.
Other stadiums across Germany planned rainbow light displays, including Berlin’s Olympic Stadium, as well as Bundesliga stadiums in Cologne, Frankfurt and Wolfsburg.
In Budapest the Portuguese face a France side who have already qualified but will want to win to secure top spot, meaning a theoretically easier tie in the next round.
“We’re guaranteed to qualify and that gives us a bit of peace of mind. From experience I’m not getting caught up in the maths, you have to respect the game,” France coach Didier Deschamps said.
Ronaldo needs just two more goals to match the all-time international scoring record of Ali Daei, who scored 109 times for Iran.
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