With the score 0-0 and 26 minutes on the clock, Sergio Ramos was given a straight red card for bringing down Umut Bulut, who was through on goal.
However, after Bulut had cancelled out Gareth Bale's opener, a superb free-kick over 30 yards from goal, the visitors fell apart, and Madrid eased to victory in the second half thanks to goals from Alvaro Arbeloa, Angel di Maria and Isco.
A down-beat Mancini rued his side's slack second-half performance in the the post-match press conference.
He said: "We prepared this game very well. In the first half we were perfect but in the second half I don't know what happened. I need to talk with my players to understand what happened, because we changed everything.
"We didn't mark in the box, we didn't attack like we did in the first half. It was strange because we had a good chance to win here.
"Real Madrid are a top side and if you give them chances like we did, we cannot win."
Despite the heavy loss, the Turkish champions remain in contention of a place in the last 16 of the Champions League, although they have to beat Juventus at home in a fortnight's time to qualify.
Madrid meanwhile recorded their fourth victory in five European games this season to secure top spot in Group B.
Ancelotti revealed he had had stern words with Ramos for getting sent off, but was pleased with the way his side reacted to going down to 10 men.
"I told him that defenders should not take risks, they should always be careful about letting forwards get behind them," he said.
"The danger is when our backs are turned, not when the ball is in front of us."
But he added: "We played very well in the second half, we played with courage and character when we went a man down."
The Madrid coach was also happy with how his team performed despite being without top scorer Cristiano Ronaldo for the first time this season.
"Without Cristiano it's quite different but his absence motivates the rest even more," he said.
The absence of Xabi Alonso, Luka Modric and Sami Khedira, who will miss the rest of the season with damaged knee ligaments, forced Ancelotti to field an untried midfield of Isco, Asier Illarramendi and Casemiro, but the Italian is confident his team's midfield is in good hands,
"Losing Khedira was a problem for us but Illarramendi was very good in Almeria, Casemiro was good today, as was Alonso when he came on, and we've got Modric coming back so I think we'll be fine without Khedira."