The goal-scorers Raheem Sterling and Gabriel Jesus took advantage of errors by defender Raphael Varane.
Real, trailing 2-1 from the first-leg and without their suspended skipper and defensive lynchpin Sergio Ramos, made a disastrous start, gifting City a ninth-minute opener.
Frenchman, Varane, having received a pass inside his own area from goalkeeper Thibaut Courtois, was robbed by City forward Jesus and he slipped the ball to Sterling who slotted home.
There was a strong response from the Spanish champions, with City keeper Ederson forced into fine saves to keep out efforts from Karim Benzema and Eden Hazard.
Ederson could do nothing, however, to keep out a bullet header from Benzema in the 28th minute, the Frenchman rising to power home a superb pinpoint cross from Rodrygo.
Man City created chances after the break but lacked the killer touch and it took another piece of calamitous defending by Varane to decide the tie in the 68th minute.
This was after his attempted header back to Courtois was too weak and Jesus was alert again, nipping in to flick it past the keeper and make it 2-1 on the night.
