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Diaz struck in the 89th minute and Olise in stoppage-time to seal treble-chasing Bayern's place in the semi-finals, where it will play holder Paris Saint-Germain.

Madrid had substitute Eduardo Camavinga sent off in the 86th after two bookings in eight minutes before its double scorer Arda Guler got his marching orders for dissent at the end of the game as the frustration of the record 15-times European champion boiled over.

It took only 35 seconds of a pulsating five-goal first half for Real to score, with 40-year-old goalkeeper Manuel Neuer, man of the match in last week's first leg, making a huge blunder and sending his forward pass straight to Guler who fired into an empty goal.

The Bavarian side, fresh from setting a new Bundesliga record for most goals scored in a single season, quickly made amends five minutes later courtesy of unmarked Aleksandar Pavlovic' header close to the goalline with keeper Andriy Lunin completely out of position.

The 21-year-old Guler came to the rescue for Madrid again, whipping a 29th minute free kick over the wall past Neuer and into the top corner to put it 2-1 up.

Guler had not scored a goal in his previous 20 matches in the competition.

Bayern bounced back again when Kane netted his 12th goal in this season's competition eight minutes later, before Madrid, battling for its only realistic shot at a trophy this season, responded through Mbappe, who levelled from a Vinicius Junior assist just before half-time.

The pace dropped after the break but Neuer, making a record-equaling 25th appearance in the UEFA Champions League quarter-finals, had to come to the rescue for the hosts, blocking Mbappe's powerful shot early in the second half.

But there was still more drama left in the game with Cavaminga sent off four minutes from the end for holding on to the ball after a foul.

Bayern took full advantage of the extra man, scoring through Luiz's deflected strike before man-of-the-match Olise crowned a famous win for the six-times European champion with a stunning late curler with the last kick of the match.

“It was all over after the red card," Real coach Alvaro Arbeloa said.

"You can’t send a player off for something like that on such big stage. It´s a shame to spoil such a beautiful game that we were all enjoying, an amazing night that was directly impacted by the referee decision."

Kane, meanwhile, called on Bayern to push on after such this dramatic success kept its treble hopes alive.

With a blockbuster meeting with Paris Saint-Germain awaiting Bayern in the last four, Kane says he and his team-mates will relish the coming weeks.

"When you beat a team like Madrid, especially in the later stages of the Champions League, it will give you confidence," Kane said.

"But we have spoken about this all year, being at this stage and being in all the competitions... this is the time to push on. 

"We feel we can beat anyone when at our highest level and we'll need that in the next round because PSG are one of the best teams in Europe.

"We have a chance to win the league this weekend, so hopefully we can get that done and we have a big semi-final in the cup. There are plenty of big games plenty to play for."

Madrid was enraged by the decision to send Camavinga off, but Kane said it could have no complaints after the midfielder picked up the ball to prevent a quick restart.

"There were decisions that went against us the whole game and the rules are the rules," he said when asked about the red card.

"It was a clear yellow card, I've had a couple of those in my career that have affected me. The referee has to make a decision there, and it went our way today."

Olise, who sauntered inside from the right flank before finding the top-left corner for his goal, echoed those sentiments and suggested Bayern would have gone on to win the tie regardless of the sending off.

"It was a good match, back and forth," he said. "We knew they would try to come back. At first, it was a bit like goal, goal, goal, but eventually when it calmed down, we played a bit better in the second half and we took our chances.

"The red card was late on. We had built momentum before that, the game was going our way. Maybe it helped a bit, but the game was still going how it was going."

Bayern faces Stuttgart in the Bundesliga next up, and could win the title depending on results elsewhere, before facing Bayer Leverkusen in the DFB-Pokal semi-finals next week.