The Cameroon defender took advantage of sloppy marking to find space in the box and steer home Wesley Sneijder's inswinging corner on 65 minutes to leave the tie evenly poised.
Fernando Torres had silenced a raucous home crowd when he fired Chelsea into a ninth-minute lead but the Premier League leaders could not press home their first-half dominance and the Turkish side improved markedly after the break.
Galatasaray manager Roberto Mancini, who won the Premier League in 2012 with Manchester City, said his side had deserved at least a draw.
"Maybe in the first half we were too deep, we played with fear probably," the Italian told reporters.
"In the second half we probably deserved to score another goal because we played very well. 1-1 is a good result, we have another game...
"I think my players saw what we are capable of, and we now know we can actually go through."
His opposite number Jose Mourinho, who replaced Mancini as Inter Milan coach in 2008 and subsequently guided the Italian club to the Champions League title in 2010, was also satisfied.
"This is Turkish football, it's very emotional, Galatasaray are a big club with big players and it was not easy," the Portuguese said.
"In the second half they put pressure on us, they had attacking full-backs and powerful strikers. It's better to be 1-1 than 0-0."
Drogba, Chelsea's record European scorer with 34 goals in 69 games, was kept quiet in his first game against his former side since scoring the winning penalty when the Londoners lifted the Champions League trophy in May 2012.
Closely watched by Gary Cahill and John Terry throughout, Ivorian Drogba did escape their shackles for one brief moment and it should have led to a goal for the hosts.
Drogba headed a free kick down towards the far post but Selcuk Inan was guilty of a bad miss from close range as he could only steer his effort against the outside of the post.
Having enjoyed the letoff, Chelsea F.C. fell asleep a minute later at a corner when no one picked up Chedjou and the defender gleefully equalised from six metres out.
Chelsea F.C., always dangerous on the counter-attack, had controlled the opening 45 minutes but just had Torres's early goal as reward.
Andre Schuerrle set the overlapping Cezar Azpilicueta free on the left flank and his low cross was met by the Spaniard to fire low into an unguarded net.
The hosts countered with Izet Hajrovic blazing over Sneijder's cutback and from another rapid Chelsea F.C. break, Ramires, with his bloodied nose plastered from an earlier aerial challenge, fired a Schuerrle cross high and wide.
Torres was denied a second by a sprawling save from Fernando Muslera soon after the restart but the game then went through a scrappy period, punctuated by a flurry of yellow cards shown by Spanish referee Carlos Velasco Carballo.
With their tails up following Chedjou's equaliser, Galatasaray pressed forward and Alex Telles forced Petr Cech to turn over his stinging drive but the Premier League club stood firm.
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