In the other game Real Madrid strolled to a 6-1 win in Germany against Schalke 04.
Chelsea led at half-time thanks to an early goal from Fernando Torres, but Galatasaray improved in the second half and restored parity after the hour mark when Cameroon defender Aurelien Chedjou found the net.
A draw was a fair reflection of the way the game went, but Chelsea will now hope to finish the job in the return at Stamford Bridge on March 18.
In the other game Real Madrid pounced on Schalke 04 mistakes as they romped to another 6-1 away win in the Champions League on Wednesday, their second of the season, leaving the ragged Bundesliga side with a huge gap to cover.
Karim Benzema and Gareth Bale both scored braces with Cristiano Ronaldo marking his return from a three game suspension with a double as well.
Incredibly, the home fans kept singing until the end and were rewarded when Dutch striker Klaas-Jan Huntelaar volleyed in from the edge of the area for a stoppage-time consolation.