Felipe Melo, Wesley Sneijder and Didier Drogba all found the net in a largely one-sided match between two sides trying to hang on the coat-tails of Real Madrid and Juventus in Group B.
Such was Galatasaray's dominance that it could, and perhaps should, have been more, but Roberto Mancini's men looked to take their foot off the pedal in the second half with the points secure.
There was time for Domingues Claudemir to get a late consolation for the visitors, but it will bring little comfort on the journey home.
Sneijder had already warmed the palms of Johan Wiland when Melo put the hosts ahead nine minutes in, rising to head home Dany Nounkeu's cross from the right.
Drogba then tested the FCK goalkeeper twice in quick succession, but mis-hit his first effort before forcing a better save with the second.
The former Chelsea man then tried again from long-range with a 29th-minute free-kick, but the goalkeeper made it look comfortable enough.
The FCK forwards were feeding on scraps, and Nicolai Jorgensen curled a rare chance wide from outside the area in the 35th minute.
Moments later, his side trailed by two goals as Emmanuel Eboue teed up Sneijder to slot the ball home.
And Galatasaray then killed the game off before half-time as Drogba scored in stoppage time, finishing another Eboue cross from close range.
Mancini seemed to call off the hounds in the second half as the chances dried up.
The hour mark was approaching by the time Wiland tipped over a close-range shot from Sneijder, and Burak Yilmaz then missed the target twice when presented with decent openings.
As the game moved into the final 10 minutes, FCK began to show a little threat, with Fanendo Adi and Jorgensen both enjoying openings, and they got their goal in the 88th minute as Claudemir hit a low shot into the right-hand corner of the net.