Wednesday's comprehensive win in the Bernabeu has left Real overwhelming favourites to book a third straight semi-final appearance in the Champions League.
But Mourinho says he will not be following the Spanish media's lead and writing off Galatasaray ahead of Tuesday's second leg.
"The result is very good," he said, "but in football I have seen so many incredible things and nothing can surprise me. Istanbul is a difficult place to play.
"They have players with a lot of experience, great character and personality and they will fight very hard against us. We know we have to go there very seriously as that is the only way to get through.
"I think we played seriously in the first leg. The Spanish press are not so respectful to Galatasaray but we are respectful of them."
Real kept a clean sheet at home for the first time in this season's competition, and Mourinho said: "We knew they had a good team, they had fantastic strikers, they had danger in every attacking action and we prepared very well for the match by the defensive point of view - we defended better than we attacked.
"In the attacking third I think we lost the last ball, the last pass we had easy transitions - 3 on 3 and 3 on 4, and we could have scored more goals, but I think by our defensive point of view we had a good analysis of the situation and the team played very well."
Mourinho also rejected suggestions second-half bookings for Xabi Alonso and Sergio Ramos, which rule both out of the return leg but would mean a clean slate for the semi-final, were earned deliberately.
He said: "I lost Xabi Alonso and I was thinking about playing Pepe in midfield in Istanbul. (Ramos') yellow card leaves me with a problem to resolve."