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Spain missed two of their spot-kicks, including one by substitute Alvaro Morata, whose late equaliser had taken the game into extra time.
The brilliant Danish midfielder was resuscitated in front of shocked fans and a massive global TV audience during his nation's first match of the tournament.
Captain Kane scored a brace as England crushed Ukraine 4-0 to storm into the last four.
Denmark beat the Czech Republic 2-1 in a nervy European Championship quarterfinal to reach the semifinals of the competition for the first time since they won the title in 1992.
Former England footballer Gary Lineker mocked the 31-year-old Lazio player.
Belgium were among the favourites for the tournament, having spent more than 1,000 days at the head of the FIFA rankings.
Italy set up a semifinal clash against Spain after first half goals from Nicolo Barella and Lorenzo Insigne led them to a 2-1 win.
Mikel Oyarzabal struck the winning penalty after Switzerland's Ruben Vargas fired over the bar as Spain won 3-1 in a shootout following a 1-1 draw.
The availability of both players has Belgium on tenterhooks ahead of the match in Munich on Friday after they were forced off the pitch injured in the last-16 clash with Portugal on Sunday.
Romelu Lukaku will likely be the main attacking threat when Belgium takes on Italy in Munich on Friday in the Euro 2020 quarterfinals. The Inter Milan forward has scored three goals in his team's four victories so far.
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