Bayern, the 2013 Champions League victor, had made it at least to the semifinals for the last five seasons.
Ronaldo was standing in an offside position when he fired in Real Madrid's second - and then Bayern crumbled.
But Real were pushed all the way by Bayern, who battled back from a 2-1 first-leg deficit to send the tie into extra time thanks to Robert Lewandowski's penalty and an own goal from Sergio Ramos.
Bayern Munich left the Santiago Bernabeu furious with referee Viktor Kassai and his team of assistants after being knocked out of the Champions League by Real Madrid on Tuesday. In a quarter-final you have to have a referee with more quality, I think.
"I understand the public and Cris", he said.
However, the Portuguese's second and third goals were offside, whilst visitors Bayern had Arturo Vidal sent off five minutes from the end of normal time.
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'We wanted to go through and there's a lot of anger in that a match of such intensity is decided by the referee.
Ronaldo levelled the score, heading beyond Neuer in the 76th minute from a fine cross by Casemiro, but Bayern restored overall parity within a minute.
Madrid were once again grateful to Ronaldo's killer touch as the four-time World Player of the Year was pivotal in both legs against Bayern. In trying to snuff out Lewandowski after substitute Thomas Müller had helped the ball along, the ball bounced off the Real Madrid captain and creeped into his own net. That moment in the 84th minute was a defining one for the match and 10-man Bayern found themselves no match for Madrid as play went into extra 30 minutes.
He added another from close range after a run by Marcelo down the left flank in the 110th, and Asensio sealed the victory after getting past a series of defenders and firing home from inside the area in the 112th.
"I don't say this", the Italian said.