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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.
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.
Bilbao are the fourth oldest Spanish club and among three never to be relegated, alongside Real Madrid and Barcelona.
Rubiales was returning to Spain amid a judicial probe into the business deal to hold the Spanish Super Cup in Saudi Arabia.
Indonesia's Rinov Rivaldy and Pitha Haningtyas Mentari did not have to work too hard as they outplayed the Indian duo in a contest lasting only 29 minutes.
The women's doubles pair of Tanisha Crasto and Ashwini Ponnappa losy to sixth-seeded Lee Chia Hsin and Teng Chun Hsun in the quarterfinals.
Indian men's doubles duo M R Arjun and Dhruv Kapila, mixed doubles pair B Sumeeth Reddy and N Sikki Reddy, women's doubles combo Tanisha Crasto and Ashwini Ponnappa also entered the quarterfinals.