An own goal by James Tarkowski cancelled out a disputed Brentford penalty as the west London derby finished level at the break.
The visitors had gone in front when captain Jake Bidwell was adjudged to have been brought down by Ashley Richards on the corner of the box, though the Fulham full-back appeared to take a piece of the ball.
Alan Judge sent Whites keeper Andy Lonergan the wrong way to convert the spot-kick.
A strong wind made good football very difficult and the 17th-minute penalty was the first real incident of note.
Brentford looked the better side when they got the ball down and played, with Alan Judge picking up a loose Tim Ream pass and feeding Lasse Vibe, whose shot was saved by returning Whites keeper Andy Lonergan.
At the other end, Ryan Fredericks had a shot pushed away by David Button in the Bees goal.
But Fulham put pressure on in the closing stages of the half and were rewarded when Ross McCormack’s cross, which appeared to be floating into the hands of Button, was flicked past him by the head of Tarkwoski.
Fulham: Lonergan; Richards, Stearman, Ream, Garbutt; Fredericks, O’Hara, Parker, Cairney; Dembele, McCormack.
Subs: Lewis, Hutchinson, Burn, Hyndman, Kacaniklic, Smith, Woodrow.
Brentford: Button; Yennaris, Tarkowski, O’Connell, Bidwell; Kerschbaumer, Diagouraga; Woods, Canos, Judge; Vibe.
Subs: Bonham, Barbet, McEachran, Saunders, Jota, Swift, Hofmann.