Brimming with confidence in front of 50,000 passionate fans, the Turkish champions made a blistering start, going ahead through a stylish strike from Burak Yilmaz who became the competition's joint top scorer with seven goals.
But a goal on the counter-attack from Schalke's Jermaine Jones just before half-time injected a dose of realism into the Istanbul club, who won the UEFA Cup in 2000 and were Champions League quarter-finalists in 2001.
Drogba struck up a good understanding with Yilmaz, but Galatasaray coach Fatih Terim complained that along with Dutch playmaker Sneijder they were not working hard enough.
"At the moment we are unable to apply the correct pressure on our opponents because Sneijder, Drogba and Burak are not pressing enough up front," a frustrated Terim told reporters.
Sneijder, who joined from Inter Milan last month, was substituted after the first half, clearly struggling for match fitness.
The Dutchman helped Inter to a Champions League, Serie A and Italian Cup treble in 2010 and played in the Netherlands team who finished second in the World Cup. Since then he has struggled to maintain form and was embroiled in a contract dispute in Milan.
Galatasaray's optimism has also been boosted by their strong position domestically, leading the first division by six points ahead of Istanbul rivals Fenerbahce and fired by Yilmaz's 12 goals.
But the club are under financial pressure to progress to the Champions League last eight because of the big money paid for Drogba and Sneijder.
Drogba's deal included transfer fees totalling six million euros and payments of 15,000 euros per match, plus signing fees of four million euros.
Drogba, who scored the equaliser for Chelsea in their 1-1 draw with Bayern Munich in last year's final before netting the decisive penalty in the shootout, moved to Shanghai Shenhua but cut short his stay in China amid reports of unpaid wages.
Schalke are considering whether to appeal to UEFA over Drogba's participation in the match, given controversy over his departure from Shanghai.
Sneijder signed a three-and-a-half-year contract with Galatasaray and will cost the club around four million euros a year.
Galatasaray shares are traded in Istanbul and Turkey's Capital Markets Board recently denied the company permission to make a capital increase through a rights issue, increasing the importance of Champions League revenues.
The hosts also hit the woodwork against the German side for the seventh time in seven Champions League games when former Schalke player Hamit Altintop struck the crossbar in the 18th minute.
"I am sorry about our game, we couldn't do what we wanted on the pitch. We could have scored a second goal before conceding. Schalke played better than we expected, and we were unable to achieve what the coach demanded from us," Altintop said.
"We were closed up in defence too often. The first game was difficult for us, but we are looking forward to the second game. We
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