The Barclays Premier League champions are on the brink of elimination from Europe's top club competition after crashing to a shock 2-1 defeat to the Russian side at the Etihad Stadium.
City had both Fernandinho and Yaya Toure sent off on a miserable night that ended with them bottom of Group E with just two points from four games
They will now go out if they fail to beat Bayern Munich on matchday five and there is a winner in the game between CSKA and Roma.
Manuel Pellegrini's team were held to a 2-2 draw by CSKA in Moscow a fortnight ago but, buoyed by a derby victory over Manchester United last weekend, had been confident of kickstarting their stale campaign.
Pre-match odds of 9/1 for a CSKA victory reflected expectation, but City never got going after conceding the first of Seydou Doumbia's two goals in the second minute.
Kompany said: "I think the result says enough
There is no way we can afford to lose points against a team we should beat every day, especially at home.
"But it happened and somehow you need to turn towards the future again
One of the good things about football is that there is always another chance, whether it is in this competition or the Premier League.
"It won't take a lot for everyone at the club to realise we need to do better.
"I am completely proud of how we played when we were down to nine men
Anyone that saw those last 20 minutes will see everyone gave everything they had
You can't debate that.
"But before that it wasn't good enough
It is another 70 minutes where we should have done better."
Toure was guilty of failing to pick up Doumbia as he leapt to meet Bebras Natcho's cross for the opener
He made amends with a brilliant free-kick equaliser after seven minutes, but CSKA were back ahead again before half-time as Doumbia capitalised on poor defending.
The hosts' problems deepened as Fernandinho blocked Ahmed Musa to earn a second yellow card and their frustration was evident when Toure was dismissed for raising his hands to push away Roman Eremenko eight minutes from time.
City did have some grievances
They felt Pontus Wernbloom should have been sent off in the second half
The Swede had already been booked when he pulled Sergio Aguero back to halt an attack
City complained and Sergei Ignashevich was booked amid the confusion in what appeared a case of mistaken identity
It may have been, however, that Ignashevich was actually cautioned for dissent while Greek referee Tasos Sidiropoulos did not deem Wernbloom's foul worthy of further punishment
City were also frustrated as Aguero had two late penalty appeals turned down
Kompany did not make excuses but, after three good spot-kick claims were rejected in the Manchester derby, felt City were due a change of luck.
He said: "I am not even going to talk about the referee, but at some stage I do hope it will turn to our advantage.
"I do believe a bit in karma
We have been unlucky now and eventually in the season it has to go our way
Giving the wrong man a yellow card - it's laughable.
"Okay it happened, we all make mistakes and we probably made more than the referee, but eventually I hope we are going to at least get something, where people are going to be angry a decision has gone for us."
City's form has actually been patchy for a number of weeks now, with the derby result being their only win in five games.
After beating Tottenham 4-1 on October 18, City failed to win any of their next three games with the draw in Moscow followed by defeats to West Ham and Newcastle.
Their title defence has so far not been a convincing one and the pressure will be on as they travel to struggling QPR on Saturday.
Striker Stevan Jovetic has warned the team need to improve quickly.
The Montenegro international said: "We need to forget this as soon as possible because we have the next game against QPR, which we need to win.
"We don't have much time to think
We need to play well in the Premier League and then for the next game in the Champions League we need to improve.
"We have Bayern Munich at home and then Roma away
We are a big team and we believe if we win two games we will go (through)."
Source : PA
Source: PA