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City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat.
PSG will face Borussia Dortmund in the semifinals after the Germans beat Atletico Madrid.
PSG came into the match unbeaten in their last 27 games in all competitions but were undone by a fired-up Barcelona side looking to return the club to the summit of European football.
Islamic State was alleged to have threatened four venues for this week's Champions League matches, including the Parc des Princes in Paris and two games in Madrid hosted by Real Madrid and Atletico Madrid.
Ten cities across Germany - Berlin, Cologne, Dortmund, Duesseldorf, Frankfurt, Gelsenkirchen, Hamburg, Leipzig, Munich and Stuttgart - will stage the matches.
Paris St Germain, who are seeking their first Champions League title, will take on Barcelona while Atletico Madrid face Bundesliga side Borussia Dortmund, which means all three Spanish teams have the opportunity to reach the semifinals.
Atletico Madrid needed a penalty shootout to edge past Inter Milan and reach the Champions league quarterfinals
Bayern CEO Jan-Christian Dreesen said there would a 'a new direction' next season with a new coach.
All four of their previous continental finals had finished 0-0, with them winning on penalties in 1992 and 2015, both times against Ghana, and losing the other two.
Zaire-Emery became the youngest ever French goalscorer in the Champions League at 17 years and 280 days.