Arda Guler pounced on a loose Manuel Neuer pass to put the visitors ahead after just 34 seconds and scored again from a free-kick after Aleksandar Pavlovic equalised.
Harry Kane put Bayern ahead in the tie only for Kylian Mbappe to restore parity overall when he put Madrid 3-2 up before half-time.
Eduardo Camavinga was sent off for a second yellow with four minutes left and Bayern pushed forward, Diaz blasting into the corner from outside the box after a crucial deflection.
With Real pressing for an equaliser, Bayern broke and Olise curled in a magnificent shot to rubberstamp their tickets to the last four, where Luis Enrique’s reigning European champions await.
Tempers boiled over after the final whistle with Guler picking up a straight red for confronting the referee.
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For the first time in Real’s long Champions League history, their starting XI did not contain a single Spanish player. Jude Bellingham, who impressed off the bench in the first leg, was one of four changes to Alvaro Arbeloa’s line-up for the visitors.
Neuer, widely lauded after a vintage performance in Madrid, gifted Real an opener. The Bayern goalkeeper miscued a pass directly to Guler, who floated a first-touch shot into the unguarded goal in the first minute.
Bayern looked stunned but struck back almost immediately when Pavlovic headed in a Joshua Kimmich corner after Real goalkeeper Andriy Lunin failed to read the flight of the ball.
The match had barely time to settle before Real were ahead once more thanks to a Guler goal, with Neuer again not at his best.
The Turkey international whipped a free-kick into the top corner which Neuer got a hand to but was unable to keep out.
The match continued to seesaw before half-time, with both Kane and then Mbappe getting on the scoresheet.
Kane slotted clinically into the bottom corner in the 38th minute to again haul Bayern level on the night — and ahead in the tie — before Mbappe ran onto a Vinicius Junior pass and slotted home just before the break to level the tie 4-4 on aggregate.
With Real regularly cutting into Bayern’s high line, Kompany responded by introducing the pace of Alphonso Davies at the interval.
Both sides traded chances in the second half, with Olise particularly dangerous, forcing a fingertip save from Lunin with 20 minutes left.
Camavinga came on midway through the second half but picked up two yellow cards in quick succession to leave his team a man down in the closing stages.
It proved a turning point as Bayern struck three minutes later when Diaz’s effort from outside the box took a touch off Eder Militao and flashed beyond Lunin.
Olise made certain of Bayern’s progress deep into stoppage time as the German giants took down Real in a knockout clash for the first time since 2012.
AFP
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