The Ivory Coast striker, last year's Champions League hero for Chelsea who moved to the Istanbul club last month after a spell in China, linked up well with Burak Yilmaz in attack.
Yilmaz scored his seventh goal in seven Champions League matches, putting Galatasaray ahead in the 12th minute when he picked up a pass from Selcuk Inan and flicked the ball around a defender to power in a shot past keeper Timo Hildebrand.
"We scored a beautiful goal early in the game with Selcuk's intelligent pass and Burak's talented shot. We could have scored a second goal, we found the chances, the game could have taken a totally different shape," said Galatasaray coach Fatih Terim.
Schalke equalised in the closing seconds of a pulsating first half. The visitors broke quickly, Jefferson Farfan passing across the goal to Jermaine Jones who side-footed past Fernando Muslera, leaving the German side well placed ahead of the second leg.
"We played very well tonight and did many right things on the pitch. We found many chances but made individual mistakes, one of which cost us a goal," said Schalke coach Jens Keller.
"We could have scored more goals, but I am very happy with our performance in this great atmosphere," he added. "We will learn lessons from this game and take necessary precautions for the second leg."
Dutch playmaker Wesley Sneijder, another big-name acquisition by ambitious Galatasaray, was substituted after the first half in an apparent tactical move by a player who still did not appear match fit.
Schalke, in poor form in the Bundesliga, began strongly with Jones cutting in from the left and firing in a shot, forcing a save from Muslera in the first minute.
But spurred on by the passionate home fans, Galatasaray soon began to demonstrate their attacking power in what was an open game from the outset.
The visitors had a chance to equalise soon after Yilmaz's opener, but Klaas-Jan Huntelaar failed to steer the ball in at the far post.
Hamit Altinop nearly doubled the home team's lead in the 18th minute with a powerful shot which ricocheted of the bar after Hildebrand saved an effort from Drogba, who took the winning penalty for Chelsea in last season's final shootout.
Yilmaz had another chance to score in the 38th minute when Drogba put him through on goal but Hildebrand blocked his shot.
The match continued to alternate attacks in the second half and Farfan had a chance to put the visitors ahead in the 55th minute but his effort was blocked by Sabri Sarioglu.
In the final stages, Galatasaray pressed harder for an equaliser while Schalke looked to hang on for a draw.
Yilmaz had a great chance to put Galatasaray back ahead in the 77th minute but he fired wide of the Schalke goal.
"We could have scored or conceded in the second half. Of course Schalke holds the advantage now, having forced us to a draw, but we will keep chasing the chances until the very last second," Terim said.
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