Fulham's Emile Smith Rowe, right, and Brentford's Keane Lewis-Potter battle for the ball during the English Premier League soccer match between Brentford and Fulham in Brentford. AP
The Bees could have climbed to within two points of the Champions League places as they aim to qualify for Europe for the first time.
But they only really threatened to make the breakthrough in a frantic finale as Dango Ouattara's effort was brilliantly saved by Bernd Leno.
A point does little for Fulham's own European ambitions as they remain three points behind their opponents, down in 12th.
Brentford move level on points with sixth-placed Chelsea, who host Manchester United later on Saturday, but remain behind the Blues on goal difference.
Keith Andrews' side are now six games unbeaten in the Premier League, but five consecutive draws has stalled their charge for an unlikely place in the Champions League.
The lack of resources at Andrews' disposal compared to their competitors at the top of the table was shown as the Irishman did not make a single substitution as his side went in search of a winner.
Igor Thiago was unable to move level with Manchester City's Erling Haaland in the race for the Golden Boot as Fulham kept the Brazil international in check.
Ouattara always looked the most likely player to break the deadlock and only a stunning stop from Leno denied the Burkina Faso international in stoppage time.
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