Sergio Ramos was sent off for the 17th time in his Merengue career after pulling down Umut Bulut as he raced through on goal midway through the first half.
However, Real shrugged off the loss of the Spain international as Gareth Bale fired in a wonderful free-kick to give the hosts the lead with his fifth goal in six games. Bulut levelled a minute later for Galatasaray, but second-half strikes from Alvaro Arbeloa, Angel di Maria and Isco sealed a comfortable 4-1 win for the hosts.
And Ancelotti was delighted with his side's response, particularly as they were also missing the injured Cristiano Ronaldo.
"Things were complicated by Sergio's red card, but I hope he is free to play in the last 16," said the Italian. "I think we played very well in the second half when we had a player less.
"We played with courage, personality and very well generally. To play with ten men is not always easy, but all the players worked together. Without Cristiano it is quite different, but his absence motivates the rest to do more."
Real have now scored 26 goals in their last six games as they have found their best form since Ancelotti took charge, and the former AC Milan coach is keen to keep that run going until the winter break in just over three weeks' time.
"Right now the team are in very good form," he said. "We are scoring a lot of goals, playing well, and we need to continue like this. I am very happy and I can't ask for any more."