The former Chelsea striker joined the Turkish giants from Shanghai Shenhua in January and was making his European debut in his first competitive start for his new employers.
Drogba, who won the Champions League last season, was involved in the build-up to Burak Yilmaz's seventh goal of the 2012/13 competition in the first half.
But Schalke hit back with a counter-attack on the stroke of half-time through Jermaine Jones to take an important away goal back to Germany for the second leg on 12th March.
Another Galatasaray new-boy, Wesley Sneijder, also made his return to the Champions League after completing a move from Inter Milan but he was substituted off at half-time.
It was Yilmaz who produced a moment of magic worthy of either Sneijder or Drogba when he broke the deadlock in sparkling fashion.
The forward collected the ball from Selcuk Inan on the edge of the area before flicking over Schalke defender Benedikt Howedes with his heel and wonderfully firing into the roof of the net in the 12th minute.
Both sides seemed to view attack as the best form of defence and Drogba, who was making a nuisance of himself in Schalke's backline, played in Yilmaz but he could not lift his effort over Timo Hildebrand.
But Schalke drew level on the stroke of half-time courtesy of United States international Jones, who will miss the second leg after picking up a booking for dissent, after linking up with Jefferson Farfan in a counter-attack.
It was a far more subdued affair after the interval, although Schalke did have a decent penalty appeal turned down when Farfan's fierce strike looked to have cannoned off Sabri Sarioglu's hands, which were raised to his chest.
And Gala's efforts to push for a winner nearly backfired completely when, at the other end, Julian Draxler dribbled to the edge of their box unchecked before shooting inches wide of Fernando Muslera's right-hand post.