Cottagers boss Roy Hodgson opted to rest several key players, with the likes of Andrew Johnson, Bobby Zamora, Brede Hangeland and Danny Murphy all left out of the squad.
Only goalkeeper David Stockdale and defender John Pantsil survived from the side that defeated Everton 2-1 in the Premier League on Sunday.
The wholesale changes clearly had a negative effect as the first half was a dour affair with neither side able to fashion a clear goalscoring opportunity.
CSKA dominated possession but the closest either side came to breaking the deadlock was a shot from Simon Davies on the stroke of half-time which was straight at goalkeeper Ivan Karadzhov.
The home side made a couple of changes in the early stages of the second half in a bid to inject some energy into their lethargic play.
The changes clearly had the desired effect as substitute Michel fired them ahead after 62 minutes.
Michel took a touch before picking out the bottom corner with a half-volley from the edge of the box after he had been teed up perfectly by Marquinhos' chest down.
But rather than build on their advantage, the home side allowed Fulham to get back on level terms just three minutes later.
Diomansy Kamara raced on to Erik Nevland's long ball over the top and the striker rounded a half-hearted challenge from the advancing keeper before finding the back of the net from a tight angle.
It was a quality finish from the striker who looked lively throughout a low-key match.
The game then opened up and Yordan Todorov had the chance to strike back soon after but snatched at his shot from inside the box and it flew harmlessly wide.
Nevland then repeated the trick at the other end as he scuffed a shot from right of goal and did not trouble the home keeper.
Stockdale produced a fine reaction save to deny Todorov on 77 minutes when he tipped the striker's first-time shot from six yards over the crossbar.
Fulham hearts were in their mouths in the closing minutes when a wicked free-kick was almost headed into the visitors' net by one of a posse of defenders attempting to clear the danger.
But the Cottagers held on to secure a valuable point which could prove handy when the group moves to its conclusion in the coming weeks.