Elliott delivered his second goal of the season in the 59th minute, nine minutes after bobby Zamora's seventh of the season had put the visitors ahead in a game where, for long periods, they were the better team.
Burnley v Everton previewHowever the arrival of substitute David Nugent to spark new light into the home ranks and Elliott's equaliser was the reward for an unstinting second half fight-back.
Fulham, who had not beaten Burnley on their own ground since 1951 were clearly in a hurry to improve that record and could have been comfortably ahead within the first 10 minutes.
Indeed, the best opportunities came in the opening two minutes when the home defence was all too easily exposed. In the first minute Damien Duff drifted in unopposed from the right and as Brian Jensen, the Burnley goalkeeper, advanced the former Blackburn winger chipped carelessly wide. Sixty seconds later Eric Nevland's powerful drive was beaten away by Jensen who followed up by denying Duff as Burnley continued to falter at the back.
Burnley's response had been a stinging Robbie Blake attempt that warmed the hands of the Fulham goalkeeper, Mark Schwarzer.
There remained increasing uncertainty in the Burnley defence as Chris Baird shot over although Chris Eagles provided home relief when his deflected shot from the right was taken low down by Schwarzer.
However, Fulham maintained their authority as Clint Dempsey shot over, a Baird header was saved by Jensen and an opening half dominated by the visitors concluded with Eagles shot straight into the hands of Schwarzer.
Burnley threatened on the resumption but Elliott wretchedly shot wide when set up by Andre Bikey in the 49th minute. And they were punished a minute later when Zamora struck, out-pacing Steven Caldwell when Baird drove long and high through the middle.
That lead could have doubled had not Clarke Carlisle cleared off the line when Dempsey headed in from Duff's corner. However, in the 59th minute rejuvenated Burnley struck back. Stephen Jordan crossed from the left, Aaron Hughes saw his attempted headed clearance fall into the path of Elliott who finished with aplomb.