The Russians were 3-0 down after half-an-hour and were up against it from the moment City striker Sergio Aguero converted a third-minute penalty.
Alvaro Negredo also scored a hat-trick for City, while Seydou Doumbia scored a goal in each half as CSKA did expose deficiencies in the home defence.
CSKA were beaten 3-0 by Bayern Munich in their first game of the campaign in Group D and Slutsky compared City unfavourably to the UEFA Champions League holders.
He said: "We made a lot of mistakes. Obviously we were playing against players of a high level, but those mistakes were our fault.
"We lost because of ourselves, while in Bavaria it was mostly down to the level of the other side.
"How can you be psychologically ready when after the first minute you concede a goal and then three goals during the first half?
"There was a lot of freedom to attack but we were unbalanced in defence.
"Obviously when we had chances we scored but we should be quite alert. With such a tough team as Man City you have to be alert because they can be fatal in a split-second."