The Turkish Super Lig champions won their first match in eight on Saturday, with Wesley Sneijder's double giving them a badly needed 2-1 home win over a 10-man Karabukspor.
The result lifted them to 10th in the table, and new manager Roberto Mancini is sure to demand more from his squad as the season goes on, starting with their Group B UEFA Champions League encounter against the Danes.
Sneijder has scored just seven goals since moving from Inter in January, and president Unal Aysal said it was about time the Dutchman started performing.
"Sneijder produced the type of performance (against Karabukspor) that we expected from him for the first time," Aysal said.
"Naturally, we have high expectations of him. We have waited for this for months."
The match - which will be Galatasaray's 250th in UEFA competition - will give them a chance to respond to a dreadful performance against Real Madrid in their last home outing in the Champions League.
Galatasaray were embarrassed as Cristiano Ronaldo scored a hat-trick in Real Madrid's 6-1 victory in Istanbul.
They improved in their second fixture, with Umut Bulut's 88th-minute equaliser giving them a point in a 2-2 draw at Juventus.
Copenhagen have recovered from a poor start of their own, winning three of their last four matches in the Danish Superliga.
That has seen them climb to sixth in the table, although a 3-0 home triumph against Aalborg on Sunday saw them keep their first clean sheet in 18 matches.
Copenhagen only qualified the group stages of the Champions League after Galatasaray's rivals, Besiktas, were kicked out for their involvement in a match-fixing scandal.
Stale Solbakken's side lost their play-off to Lille - 2-1 on aggregate - but were placed in Group B after Besiktas were punished.
The last time they were involved in the group stages of the Champions League, Copenhagen reached the round of 16 against the odds.
A similar performance is unlikely this time around, but a surprise win at the out-of-form Galatasaray would boost their chances.
They made a bright start to their Champions League campaign, claiming a surprise 1-1 home draw against Juventus, but were beaten 4-0 at Real Madrid in their last fixture.
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