The Bulgarian side had garnered only one point from their first four group matches and this defeat ends their interest in the competition for another season although their second-half display merited a draw at least.
With AS Roma beating 2-1 Basel in the Italian capital, the Cottagers will qualify from the group if they can beat the Swiss side later this month.
The Premier League side dominated the first 45 minutes although they almost fell behind early on when Todor Timonov hit the crossbar.
When Fulham responded, Danny Murphy, making his first appearance for six weeks after recovering from a knee injury, had a great chance to break the deadlock from the penalty spot after Zoltan Gera had been upended by Ivan Ivanov.
However, the former Liverpool midfielder side-footed his effort wide.
In the 15th minute Fulham did move ahead though when Hungarian Gera headed home Bjorn Helge Riise's cross.
Bobby Zamora missed a good chance to double the advantage before the break and the former West Ham United striker was again off-target early in the second period.
CSKA finished the game strongly though with Spas Delev firing wide after being set up by Yordan Minev while Kostadin Stoyanov was off-target with a close-range header when he should really have scored.
But the nearest they came to grabbing an equaliser came in the closing stages when substitute Damien Duff was forced to clear Kosta Yanev's drive off the line.