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City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat.
Xabi Alonso's team, who have now gone a league-record 29 matches without defeat this season to secure their first trophy since 1993, have a 16-point lead over Bayern with five matches remaining.
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.
Federico Valverde salvaged a draw for Real late on after City had taken a 3-2 lead with sumptuous strikes from Phil Foden and Josko Gvardiol following a Ruben Dias own goal and Rodrygo strike that gave the hosts a half-time lead in Spain's capital.
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.
Paris St Germain's Kylian Mbappe scored twice to lead them into the Champions League quarterfinals.
Bayern CEO Jan-Christian Dreesen said there would a 'a new direction' next season with a new coach.
The attacking full-back and dead-ball specialist became an instant Germany football great when his spot-kick earned them a 1-0 victory over Diego Maradona's Argentina in the World Cup final in Rome in 1990.