Zinedine Zidane's son Enzo started the match for Real but was quiet and subbed after an hour and instead it was Jamie Sanchez who headed home the only goal of the one leg tie in the 72nd minute.
PSG were at home for the clash in Charlety-Paris and looked to have a huge advantage when Héctor Martínez was sent-off for the visitors in the 55th minute for a professional foul.
PSG had marginally the better of things even before that with Kingsley Coman hitting the post.
However, Sánchez's header was then deflected into the net which means Real will face Benfica or Manchester City in the last four in Nyon, Switzerland.
The other quarter-finals are Chelsea v Schalke and Barcelona v Arsenal.
The UEFA Youth League comprises a group stage and four knockout rounds played as one-off games.
Access to the competition is reserved for the youth teams of the 32 participating clubs in the UEFA Champions League.
The semi-finals and final will be contested in Nyon. There is no third-place play-off.
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