Real rarely looked in any real danger and were two goals up before the half-hour mark through Cristiano Ronaldo and Karim Benzema.
A second-half header from Gonzalo Higuain sealed the more than comfortable win for Real, who have left Galatasaray with it all to do ahead of next week's second leg in Istanbul.
Mourinho has never lost a European quarter-final in his managerial history, and that record never looked in danger at the Santiago Bernabeu.
The visitors started off reasonably well, and they played an attacking formation with Wesley Sneijder, Hamit Altintop and Selcuk Inan all pushing forward to try and join Didier Drogba and Burak Yilmaz.
But, just nine minutes in and the deadlock was broken. A flowing move saw Mesut Ozil feed Ronaldo and he dinked the ball over Fernando Muslera in typically clinical fashion.
Despite going behind, the Turkish giants continued to push forward. Drogba had a decent effort pushed wide by Diego Lopez before Emmanuel Eboue then wasted a great chance after being played through by his fellow countryman - but he fired straight at the Real keeper.
Just moments later and Gala were punished for some slack defending, Michael Essien - playing at right-back - put a good ball into the box. Semih Kaya and Eboue failed to deal with it and, before they could react, Benzema had controlled and swept the ball into the net.
With Sneijder withdrawn at half-time, the Turkish side were a more solid outfit - but what they gained in defence, they lost in attack as they posed little threat in the second half.
Indeed, it was little surprise that Real extended their advantage. Xabi Alonso curled a teasing free-kick into the box and Higuain headed into the corner to score and surely end any hopes Galatasaray had of a second-leg comeback.
The visitors did have late calls for a penalty denied when Sergio Ramos stood on Yilmaz's foot - but the striker was harshly booked for a dive instead. Ramos later picked up a booking that ruled him out of the second-leg when he kicked the ball away with just minutes left.
Another real positive for Mourinho will be the fact that this was Real's first clean sheet in the Champions League this season.