The tie exploded into life after just 35 seconds. Manuel Neuer was caught out of his area, gifting possession to Arda Guler, who struck from long range into an empty net to cancel Bayern's first-leg advantage instantly.

Bayern responded within five minutes. Joshua Kimmich delivered a dangerous corner and Aleksandar Pavlovic rose highest to head past Andriy Lunin. The hosts controlled possession, but Madrid remained dangerous, with Kylian Mbappe denied by a crucial block from Konrad Laimer.

Guler struck again midway through the half, curling a free-kick into the top corner, but Bayern refused to fold. Dayot Upamecano threaded a precise pass to Kane, who calmly finished to restore Bayern's aggregate lead and mark his 50th goal of the season.

Madrid surged again before the break. Vinicius Junior rattled the crossbar before racing clear and setting up Mbappe to score, handing the visitors their third lead of a breathless first half.

The second half followed a similar pattern. Bayern pushed forward, with Diaz and Pavlovic both going close, while Neuer redeemed his early error by saving from Mbappe. At the other end, Lunin produced key stops to deny Michael Olise and others as Madrid sat deeper and looked to counter.

The decisive moment came late. With four minutes remaining, Eduardo Camavinga was sent off for a second booking, and Bayern took full advantage. Diaz combined brilliantly with Jamal Musiala before his strike deflected in to level the tie on the night and swing the aggregate.

Moments later, Olise curled home with the final kick to seal victory.

"I think even in the last 20 minutes I felt like we were the team trying to make something happen," Kane said. "They started to tire, and sometimes it just takes a moment of magic. In this case, it was two great finishes by Lucho and Michael to finish the game."

Kane pointed to Bayern's composure as the key difference. "The highs and lows, especially in the first half, were pretty unique," he admitted. "But we stayed patient. We knew as the game went on, we'd get stronger physically, and that's exactly what happened."

Bayern manager Vincent Kompany added: "We believed until the very end and never gave up."

Bayern, now unbeaten in 17 matches, will face Paris Saint-Germain in the semi-finals after a night of relentless drama in Munich.

Report: Bayern Munich in advanced talks with Fulham for Joao Palhinha transfer