A fantastic strike from Wade Elliott got the Clarets' Premier League push moving again as they came from behind to preserve their impressive home record.
Elliott had been without a goal since his winner against Everton 13 games ago but matched the quality of that strike with another stunning effort to kill off the prospect of a third straight defeat.
Fulham had failed to win in their last 28 visits to Turf Moor in a run stretching back to 1951.
But Roy Hodgson's side arrived as one of the form teams of the league with just one defeat in their previous nine games and when Bobby Zamora put them ahead after 51 minutes they must have been confident of re-writing history.
The Clarets had struggled to create their normal quota of chances up to that point but they refused to succumb to only a second home defeat of the campaign.
Elliott steered in the equaliser with a cracking right-foot volley nine minutes later and with a re-shuffle from manager Owen Coyle restoring their attacking threat, the Clarets were well worth the point to take them into Wednesday night's visit of Arsenal in fine fettle.
Striker David Nugent's return via the bench was a welcome one, while Steven Caldwell started the game after his one-match suspension.
That meant Kevin McDonald dropping to the bench and a return to Coyle's current preferred back four but the quartet had barely been reintroduced when Fulham should have taken an almost instant lead as Burnley were tested early on.
There were less than two minutes gone when Eric Nevland flicked the ball wide to Damien Duff who was in the clear with just Brian Jensen to beat.
The exposed Clarets' keeper had to come off his line but Duff failed to hit the target with an attempted chip as he sliced a tame effort wide.
Jensen might have been powerless then but quickly imposed himself with two fine saves to keep the slow-starting Clarets level.
Firstly, the great Dane blocked a close-range Nevland shot with his legs and then showed sharp reflexes to throw out a left hand and brilliantly palm away a fierce drive from Duff, who threatened to silence the Burnley fans who were quick to remind him of his Blackburn Rovers pedigree.
It was an uncertain opening from the Clarets but Chris Eagles got them moving forward with a fine ball forward to Steven Fletcher.
Fletcher was crowded out but when Fulham only half cleared, Robbie Blake pounced with a trademark volley which had Mark Schwarzer scrambling to save.
Eagles was Burnley's driving force with most of their best moments in the first half coming through the winger whose pace and trickery had Fulham fully stretched.
A typical run and deflected shot forced Shwarzer into another save but with Fletcher short of support and the Clarets' passing game not clicking into top gear, clear-cut chances were hard to come by before the break.
That changed less than five minutes into the second half when Elliott had the opportunity to put Burnley ahead.
This time there were Claret shirts around Fletcher when he won a long ball forward from Clarke Carlisle but when Bikey slipped a pass into Elliott the midfielder dragged his shot just wide.
And within moments Fulham had struck through Zamora. Once again Burnley were caught out by a quickly taken free-kick from Chris Baird which saw Zamora hold off Caldwell and run through to steer a shot across Jensen for his seventh goal of the season.
Clint Dempsey went close to a second with a header from a Duff corner and Coyle responded by sending on Nugent with more than half an hour to try and force an equaliser.
In fact it took only three minutes with Elliott the one to grab the plaudits with a spectacular strike to level the scores with an hour gone.
The change to a 4-4-2 formation saw Elliott pushed into a wide right position so he was perfectly placed to advance onto a half headed clearance from Stephen Jordan's cross from the opposite side and expertly guide a 15-yard volley past Shwarzer for his first goal since August.
Suddenly Burnley were on the front foot and with Turf Moor coming alive carried the fight to the visitors.
Nugent unleashed a shot on the break which Schwarzer held and Andre Bikey also worked the Australian international with a skidding effort from outside the penalty area.
Fulham weathered that storm and it took an alert dash off his line by Jensen to stop Zamora sniffing out a second from Paul Konchesky's through-ball.
But Burnley finished in positive mode, throwing men forward in search of a winner that just would not come with substitute Fernando Guerrero lifting the best chance high into the cricket field stand.
Burnley: Jensen, Mears, Jordan, Carlisle, Caldwell, Alexander, Bikey (McDonald 77), Eagles (Guerrero 86), Fletcher, Blake (Nugent 57), Elliott.Substitutes: Penny, Kalvenes, Gudjonsson, Thompson.
Fulham: Schwarzer, Konchesky, Paintsil, Hangeland, Baird, Nevland (Gera 77), Duff, Hughes, Dempsey, Zamora, Greening. Substitutes: Stockdale, Murphy, Riise, Smalling, Elm, Dikgacoi.
Referee: Mike Jones