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.
Making his point: Didier Drogba celebrates scoring for Galatasaray against FC Copenhagen
Going Dutch: Galatasaray's Wesley Sneijder enjoys scoring his goal against FC Copenhagen
Opening goal: Galatasaray's Felipe scored the first against FC Copenhagen
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.
The Dutchman extended his side's lead in the 35th minute as Emmanuel Eboue's found him and he slotted teed the ball home.
Thumbs up: Galatasaray manager Roberto Mancini
Mancini seemed to call off the hounds in the second half as the chances dried up.
Copenhagen got a consolation goal in the 88th minute as Claudemir hit a low shot into the right-hand corner of the net.
Bayern Munich continued their dominant form in this season's tournament by beating the Group D whipping boys Viktoria Plzen 5-0 in the Allianz Arena.
The reigning European champions maintained their 100 per cent record in this season's tournament with a comfortable win as Franck Ribery's penalty in the 25th minute and a 37th-minute goal from David Alaba put the hosts in control at the break.
At the double: Bayern's Franck Ribery scored two goals against Viktoria Plzen
Walk in the park: Bayern's Mario Gotze scored the fifth goal and, right, boss Pep Guardiola gives out orders
Getting the plaudits: Munich's David Alaba, far left, celebrates scoring against Viktoria Plzen
In the battle for second place in Group C, Oscar Cardozo bundled home a late equaliser for Benfica as Jorge Jesus's side salvaged a 1-1 draw with Olympiacos.
Alejandro Dominguez had given the visitors a 29th-minute lead in torrential rain at the Stadium of Light and the Greek side clung onto their lead until seven minutes from time when Cardozo struck.
Too little, too late: Oscar Cardozo, right,
tries to take the ball from Olympiakos' Jose Holebas after an equaliser
for Benfica but the match would end in a draw
Five days after he got the winner against Hoffenheim in the Bundesliga with a shot which went through a hole in the side-netting, the striker started and completed the scoring with two legitimate efforts.
He got the Germans up and running with a header midway through the first half and tapped home a fourth 19 minutes from time as they ran riot after the break.
Simon Rolfes had doubled the lead from the penalty spot shortly after half-time and Sidney Sam quickly added a third as the Ukrainians were swept aside.
Sealed with a kiss: Leverkusen's Stefan Kiessling scored two goals against Shakhtar Donetsk
Kiessling claimed on Tuesday that the criticism he had received over Friday's incident, when his 'goal' was incorrectly awarded, had been harsh and he wanted the fuss to blow over.
He played his part in helping that happen as he gave the hosts the lead with the first real chance of the match in the 22nd minute.
Son Heung-min's shot was blocked, but the ball broke to Giulio Donati, who whipped in a cross from the right which Kiessling met in the centre of the area and headed into the bottom left corner.
Home comforts: Leverkusen's Sidney Sam, left, scored the third goal against Shakthar Donetsk
If Shakhtar had hopes of mounting a second-half fightback, those were dashed within 20 minutes.
First Darijo Srna's foul on Kiessling in the 49th minute gave Rolfes the chance from the spot, which he duly took, rolling the ball into the bottom right corner.
And Sam added a third in the 57th minute, firing a low shot into the corner from inside the area after being found by Son.
Three became four in the 71st minute when Can's ball across from the right gave Kiessling a simple finish from close range.
