Solid early pressure from Chelsea rewarded them with a ninth-minute opener from Fernando Torres at the Turk Telekom Arena, but Gala controlled the game for long spells.
Mancini reshuffled his pack to tactically stifle the Londoners and the hosts gained their reward through Aurelien Chedjou's equaliser just after the hour.
The 1-1 draw means that the tie is still finely balanced heading into the return at Stamford Bridge on March 18, with an away goal giving the Premier League side a slight edge, but Mancini felt that his side could have been heading to London in a more comfortable position.
Asked by Sky Sports whether he was disappointed to be coming away on level terms, the Italian said: "Yes, because in the second half we probably deserved to score another goal.
"We gained a lot of confidence from the second half, but we had too much respect for Chelsea. Maybe we were too deep in the first half, we played with fear but I'm happy.
"One-one is a good result, we have another game."
Jose Mourinho's Blues will host Mancini's men on March 18 as they pursue Champions League glory for the second time in three seasons.