Goals from Cristiano Ronaldo and Karim Benzema set Madrid on the way to victory at the Bernabeu, and Gonzalo Higuain's headed goal extended the advantage they will take to Istanbul next week for the second leg.
The wide margin of victory left Galatasaray coach Fatih Terim fearing the worst, saying: "After losing the match 3-0 we need a miracle but in Istanbul we will try to play well and win the match."
Yet Mourinho will not take the second match lightly, the Portuguese said.
"They need a miracle but miracles happen in football, the history of football is full of miracles, why can they not dream of a miracle?" said the former Chelsea manager.
"They will give their all to try to pull off a historic result and I have enough experience to know that although I am happy with the result, the tie is not over. I have to convince my players to believe that too."
Mourinho added: "The team played well and produced a professional and balanced performance.
"We defended well and were dangerous on the counter-attack. They knew that at this stage of this competition that matches are decided by the slightest of margins, and we were concentrated and played with humility."
Mourinho recalled Benzema to his starting line-up in place of Higuain, who started against Manchester United at Old Trafford in Madrid's last Champions League match.
He was pleased that both the Frenchman and substitute Higuain, who replaced Benzema midway through the second half, were able to get on the scoresheet.
Mourinho said: "Higuain and Benzema have scored goals and that's great because that will improve their confidence but I would have been happy with their performances even if they had not scored."
The coach also rejected the notion that Madrid benefited due to Galatasaray's defensive errors, stating: "The difference was not Galatasaray, it was Real Madrid, and we deserved to win the match.
"When there is a goal so early on it changes the game. They play an attacking style of football any way, but a goal will always make for a different story. They did everything to get a different result."
Terim criticised referee Svein Oddvar Moen's performance, above all his decision to book striker Burak Yilmaz instead of awarding a penalty following a challenge from Sergio Ramos late on.
"Maybe the name Real Madrid affected Galatasaray and the referee did not have courage," Terim said.
"The referee made many mistakes. Burak was fouled, everyone saw it except for the referee, but despite everything there is nothing we can do, the result is there for everyone to see."
He also rued his side's carelessness in conceding Ronaldo and Benzema's goals, as well as pointing the finger at Didier Drogba and Emmanuel Eboue, who both missed chances to score in the first half.
He said: "At this level in these kind of matches you cannot make clear mistakes. The first and the second goals were basic mistakes and then with Eboue and Drogba we couldn't take two clear chances. If we had taken advantage of those moments the match could have been different."