Former Manchester City manager Roberto Mancini, in charge of Galatasaray for less than a month after replacing the sacked Fatih Terim, steered the Turkish side to their first win in this season's Champions League after snatching a draw against Juventus in Turin in his first game.
Galatasaray, last year's quarter-finalists, had lost their first group match at home to Real Madrid 6-1, and the Istanbul side will look for another three points when they travel to Copenhagen in a fortnight.
"We automatically stepped back and played in a more conservative way in the second half, but tonight we are delighted by the result and we will be looking forward to repeat it at the next match," defender Semih Kaya said after the game.
Galatasaray's aggressive start paid off quickly as Brazilian midfielder Felipe Melo raised the roof in the 10th minute with a powerful header after Drogba's effort to win the ball near the right corner flag allowed Dany Nounkeu to cross.
In the 38th minute Dutchman Wesley Sneijder found the back of the net with a classy right-foot shot after receiving a pass from Eboue.
Eboue was the again the creator in first-half stoppage time, allowing Drogba to score the team's third goal after finding his compatriot's feet with an incisive ball past the opposition defence.
Copenhagen pulled a goal back two minutes before full time when Claudemir poked the ball past Fernando Muslera.
Galatasaray have four points, five behind leaders Real Madrid, while Copenhagen remain bottom of the group with one point.