However, Cottagers manager Roy Hodgson will be the more disappointed boss as his team enjoyed the better of the exchanges and created more clear-cut chances.
The first of those was in the opening moments of the game when ex-Blackburn wide man Damien Duff raced clear of the Burnley defence, but tamely lobbed wide of goal when he only had goalkeeper Brian Jensen to beat.
Moments later it was the turn of striker Erik Nevland but this time Jensen produced a fine save to push the ball behind for a corner.
Duff, then created another opportunity this time striking his shot well from 20 yards only to see his effort parried by Jensen.
At the other end whilst the Fulham defence enjoyed few scares, goalkeeper Mark Schwarzer made fairly routine saves from Robbie Blake and Chris Eagles.
Burnley had a great opportunity to take the lead early in the second half but Elliott shot wide from a good position on the right-hand side of the penalty area.
This was to prove an expensive miss as shortly afterwards the visitors took the lead when Burnley were caught napping at a quickly taken free-kick which skipper Steven Caldwell failed to cut out, which enabled Zamora to take full advantage and plant the ball firmly beyond Jensen.
The lead was short lived however as Clarets left-back Stephen Jordan crossed into the box for Elliott to bring the scores level with a fine volley past Schwarzer.
But the visitors continued to look the more threatening of the two teams and almost stole the points towards the end only for Zamora to be denied by Jensen.
Zamora was again prominent in many of the good things which were undertaken by Fulham and has shown why his manager thinks that he could have an outside chance of a place in the England World Cup squad.
Turf Moor remains something of a bogey ground for Fulham who now have not won in 28 visits to the ground.
This was the first time in the season which Burnley have come from a goal down to earn a point and no doubt manager Owen Coyle will be pleased with that side of his team's efforts.
The concern will be that Burnley have not won since the end of October in a five-match winless streak and with Arsenal the visitors in midweek the likelihood remains that it will be a long hard winter for the Clarets.