Diomansy Kamara picked himself up from a Marco Andreolli foul to put Fulham ahead from the penalty spot.
Diomansy Kamara put Fulham back into it with 20 minutes left by drilling a shot through 'keeper Joe Hart's legs.
Fulham were attempting to make the most of their good fortune, and Diomansy Kamara tested Bywater after a mazy run.
But Senegal deservedly restored their lead after 47 minutes after the impressive Diomansy Kamara neatly flicked home the lively Camara's cross.
With Roma pressing, Zamora broke but his cross was turned behind by Philipe Mexes at full stretch with Diomansy Kamara waiting to pounce.
Mamadou Niang has an impressive international scoring record, Diomansy Kamara is playing regularly in Italy's Serie A for Modena, and Mamadou Diabang is on fire in the German Bundesliga.
Diomansy Kamara leveled three minutes later with his first goal of the season, as the Senegal striker calmly pushed a long ball past the onrushing goalkeeper and found the net.
Hodgson brought on Diomansy Kamara to replace Nevland on the hour and Fulham, at last, showed glimpses of creativity as Zoltan Gera, Dickson Etuhu and substitute Seol Ki-Hyeon had chances to score.
It did little to halt the hosts' dominance in the second half, but against the run of play Dempsey provided Fulham with a glimmer of hope, bundling the ball home from Diomansy Kamara's cross.
The first opportunity of the match came when a long ball from Chris Baird found Diomansy Kamara, who controlled on the edge of the six-yard box, swivelled and forced a good save low down from Carson.
It was not a day free of concerns, however - Wenger admitted his side's defence looked "vulnerable" and also revealed defender Laurent Koscielny had suffered concussion after a clash of heads with team-mate Sebastian Squillaci in the build-up to Fulham's goal, scored by Diomansy Kamara.
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