Fernando Torres netted an early away goal for Chelsea at the Turk Telecom Stadium and Mancini was forced into a tactical change after just 31 minutes, replacing forward Izet Hajrovic with Yekta Kurtulus to shore up the midfield.
Aurelien Chedjou equalised in the second half to leave the last-16 tie delicately poised, and the Galatasaray boss was relieved that his substitution helped stem the tide.
He told Sky Sports: "We conceded three or four counter-attacks and they scored. In the second half we played very well.
"Maybe in the first half we went too deep and played with fear probably, but I am happy for the second half.
"Every attack we lost four players in the first 20 minutes and we conceded a counter-attack every time, which is impossible against Chelsea.
"We needed another midfielder."
Having drawn level after 64 minutes, Galatasaray had chances to score again and Mancini was disappointed in the end not to win the game.
He said: "Yes, we probably deserved to score another goal. In the second half we played very well.
"But I think 1-1 is a good result and we have another game.
"We got a lot of confidence in the second half. We had too much respect for Chelsea in the first half."