The Cote d'Ivoire striker and Netherlands playmaker helped Roberto Mancini's men to a comfortable win in Turkey.
Galatasaray kept its Champions League hopes alive with a 3-1 win over Copenhagen on Wednesday.
First-half goals from Felipe Melo, Wesley Sneijder and Didier Drogba handed Roberto Mancini his first victory in Europe with the Turkish side, despite Claudemir's late strike.
It keeps Galatasaray within a shout of qualifying from Group B - which is led by Real Madrid and also includes Juventus - while its Danish opponents are rock-bottom with one point from three matches.
Galatasaray coach Mancini opted to make three changes from the 2-1 Super Lig win over Karabukspor on Saturday, with a double switch in the back seeing Dany Nounkeu and Emmanuel Eboue inserted. There was also a return to the starting lineup for 18-year-old attacker Bruma, who was preferred to his more senior counterpart Ceyhun Gulselam for the encounter.
There were two changes for Copenhagen boss Stale Solbakken following the 3-0 league win over AAB: Pierre Bengtsson came in for Christoffer Remmer, and Daniel Braaten was also given the nod in a more experienced lineup.
It was the home side that was quickly on top, though, with Melo putting his men ahead after just 10 minutes, heading home a stunning cross from Nounkeu that left goalkeeper Johan Wiland with little chance.
The strong start clearly fired Mancini's men up as Drogba went close to doubling the tally only minutes later, but after holding off Ragnar Sigurdsson he saw his shot blocked by Wiland.
Sneijder was on hand to fire home the second in the 38th minute, though, converting from the edge of the box after collecting a pass from Eboue.
And Galatasaray practically wrapped up the points on the stroke of halftime when Drogba added the third from inside the area.
It was little surprise that Solbakken changed things around at halftime, adding Christian Bolanos and Fanendo Adi to the mix. It looked to have paid off soon after as Nicolai Jorgensen flicked on a fine ball, only to see Aurelien Chedjou cut it out.
The visitors did get their consolation with four minutes left, but Claudemir's measured finish past Fernando Muslera was too little, too late.