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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.
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.
Rummenigge, a former longtime Bayern CEO, was West Germany's captain under coach Beckenbauer in the 1986 World Cup where they lost to Argentina in the final.
The German striker also added he would try to play cricket in his garden.
Munich: Mathys Tel's stoppage-time goal proved enough for Bayern Munich to beat Manchester United 4-3 in their Champions League opening game. Tel scored with a spectacular finish in the second minute of injury time on Wednesday, restoring Bayern's two-goal lead just when United threatened to pull
England captain Kane, who joined for a league record 100 million euros last week to solve Bayern's scoring problems, slotted their second goal in the 74th minute after setting up Leroy Sane for a fourth-minute opener.
Despite a stellar scoring record he failed to win any domestic or international titles with Spurs.