Ola Aina’s instinctive second-half strike earned Fulham a point and moved them within six points of safety.
The former Chelsea defender reacted smartly to chest the ball home from the near post after Joachim Andersen had cleverly flicked on Ademola Lookman’s corner.
But the 1-0 lead only lasted three minutes, as Burnley drew level in slightly fortuitous circumstances.
Ashley Barnes’s poor touch from a Jay Rodriguez pass fooled keeper Alphonse Areola and allowed the burly striker to net with a low finish that denied Fulham a second successive win.
That two goal-burst just after the break brought much-needed life to a dull contest at Turf Moor after a first half which brought very little noteworthy goalmouth action.
Areola was the only keeper seriously troubled when he had to push away a dangerous low cross by ex-Brentford defender James Tarkowski.
After the goals, both sides had chances as the game opened up in the closing stages, with Rodriguez forcing a decent save from Areola.
But Scott Parker’s side, who have lost just once in their past six league games, posed the greater threat, with Ruben Loftus-Cheek heading wide when under pressure and Lookman narrowly missing the target with a fine curling shot.
However, Fulham were unable to find a winner which would have taken them to within four points of fourth-from-bottom Newcastle United and had to settle for 10th Premier League draw of the season.