Apart from a worrying spell after the interval that kickstarted a lively second half, Fulham were never overwhelmed and responded superbly to Michel Platini's fine 63rd-minute opener.
CSKA were cranking up the pressure only to be caught on the break three minutes later, Erik Nevland finding Kamara who rifled home in front of 260 travelling fans for his first goal of the season.
It was a sweet strike for Kamara, who has become unsettled at Craven Cottage since slipping down the pecking order with reports linking him with a loan move to Newcastle.
Manager Roy Hodgson fielded a severely-depleted side for the Group E opener with key players such as Andy Johnson, Brede Hangeland and Danny Murphy absent.
Only goalkeeper David Stockdale and defender John Pantsil remained from the side that dispatched Everton 2-1 on Sunday, but the makeshift line-up coped admirably with the Bulgarian league leaders.
The introduction of an additional assistant referee alongside each goal passed without incident, though in a dismal first half this owed much to the lack of action in the penalty area.
Ivan Stoyanov provided one of the few highlights when he reacted smartly to some careless play by Fulham, linking neatly with Yordan Todorov before firing narrowly wide of the left post.
The teams took turns to give away possession in midfield and both lacked penetration in the face of some energetic defending.
Slowly CSKA began tightening the screw, spending more time in Fulham's half without actually threatening the goal and their period of domination ended without reward.
Simon Davies floated in a free-kick but Kamara was comfortably shackled by Ivan Ivanov and the ball flew to safety.
In a rare entertaining moment of an otherwise dull first half, Pantsil showed tenacity and skill to beat four players close to the byline before finally being dispossessed.
Christopher Smalling nodded a corner over the crossbar but then Fulham were on the defensive, Todor Yanchev picking out Stoyanov who failed to trouble Stockdale.
Jonathan Greening found space down the right, weaving his way into the area where he set up a tricky chance for Davies but the Welshman struck his shot straight at Ivan Karadzhov.
Yordan Todorov blazed wide of the near post as CSKA made an energetic start to the second half.
Slippery footwork from substitute Marquinhos was concluded with a superb flat ball into the box that CSKA claimed Stephen Kelly handled, but referee Darko Ceferin was not persuaded and Fulham escaped.
The Bulgarian club looked a different outfit after the interval, attacking with purpose and flair.
Spas Delev butchered a magnificent chance created by the influential Marquinhos, but CSKA did not have to wait much longer for a goal.
Growing increasingly ragged, Marquinhos chested the ball down to Michel Platini who buried the half-volley from the edge of the area for an impressive 63rd-minute strike that the home fans celebrated wildly.
The lead lasted just three minutes, however, as the bounce of Nevland's pass deceived Karadzhov and Kamara was able to skip away from the keeper before finding an empty net.
Stockdale produced a stunning save to deny Yordan Todorov, showing lightning reflexes to palm the ball over the crossbar.
The game continued to be played at pace but chances dried up and Fulham were able to celebrate their share of the spoils.