Ola Ania opened the scoring for the away side early in the second half in what would turn out to be a wild couple of minutes in Lancashire - ending with Ashely Barnes converting from 10 yards to make sure the gap between the Clarets and the drop zone remained eight points.
The draw - although Scott Parker will rue the chances his side missed throughout - cuts Fulham's, - what seemed like only a few weeks ago - mammoth survival effort down to just the six points. Having gained four points on Newcastle this week, a win against Sheffield United on Saturday evening could have the Whites within touching distance of exiting the relegation places.
Story of the match
Parker decided to stick with Sundays winning formula as left his side unchanged from the victory over Carlo Ancelotti's Everton side, in the hope of capitalising on his sides recent momentum .
There was little action to dissect in the opening 45 minutes, as both sides spent the large parts of the half assessing one another. With potentially huge ramifications for the loser, neither side would be blamed for not exchanging big blows so early on.
Fulham showed signs of defensive frailties early on as Burnley's notoriously tough set-pieces saw a defensive mix up between Joachim Andersen and goalkeeper Alphonse Areola - as the pair unorthodoxly tried to clear the ball away from the on-rushing Barnes.
Fulham did however, enjoy better spells of the ball throughout the first half as they examined the claret and blue defensive wall, in the hope finding any signs of a potential crack.
Kevin Long - who was brought into the side to replace the concussed Ben Mee - nearly opened the scoring just before the half-hour mark as he got his head to Ashley Westwood's corner. It was a wasted chance judging by the high aerial standards set over the course of Sean Dyche's Burnley tenure - where he ought to have done better, as his effort sailed over the crossbar.
Ambitious efforts from both sides summed up a rather mundane opening half in Lancashire as both Westwood and Mario Lemina tried their luck with volleys from outside of the area.
Parker's men looked dangerous with the ball out on the flanks with both of his wide men possessing the speed and unpredictability that gives Whites’ fans excitement every time the ball ventures into the oppositions half. Some creative play down the right left Sunday's star-man Josh Maja inches away from poking the away side into the lead as he looked to pounce on Kenny Tete's ferocious ball into the box. Only for the ever-reliable James Tarkowski to snuff the danger out.
The final chance of the half came from the least likely of sources as the board indicated how many additional minutes were to be played at Turf Moor - skipper Tarkowski combined with Dwight McNeil down the left, bursting past two white shirts as he fired in a low-driven cross across the face of Arerola's goal. Somehow the ball was missed as Burnley's two front men - Jay Rodriguez and Barnes- threw themselves at the ball in the hope of sneaking the clarets into the interval ahead.
Parker's men followed the same start to the second half as they enjoyed on Sunday evening in Liverpool, as four minutes into the restart Ola Aina opened the scoring- albeit unintentionally. As Lookman's corner was flicked on via the heel of Andersen onto the Aina's chest, as the redirection of the ball conned a helpless Robbie Brady on the line, as he sliced his attempted clearance into the back of Nick Pope's goal.
Fulham's season was summed up inside three frantic second half minutes. Having just taken the lead, Parker's men almost instantly found themselves restarting the game again in 54th minute. Some excellent play from Rodriguez as he dummied Matt Lowton's ball through the legs of Tosin Adrabioyo, giving the former Southampton man the time and space to pick out Barnes with his first touch, as Barnes made no mistake with his finish from inside the box- sitting Areola down before smashing the ball home.
There were muted celebrations from Barnes, as those in claret shirts rushed to rescue the ball from the back of Areola's net. Burnley knew if they could snatch maximum points on the night, they'd put to bed any fears of a potential relegation fight in the season's final weeks.
There were chances for either side to win the game in the dying stages as they both sought the goal that could've been worth millions come the Premier League's finale in May.
First came the onslaught from Dyche's men as Rodriguez was denied by the legs of the on loan PSG 'keeper, Areola. Substitute, Robbie Brady had the chance to steer the home side in-front with his head, but fluffing his line was all Dyche needed as the he brought an end to the Irish international's truly unforgiving cameo came to an end.
Lookman came close again in added time as he was unable to generate enough control or bend on his effort just inside the box to trouble Pope, who watched the ball sail harmless into the an empty David Fishwick stand.
On the whole it's been a positive week for Parker and his team as they massively increased their chances of avoiding an immediate return to the second tier. However, Parker and the rest of those in white shirts will be frustrated that they were unable to amplfy the pressure on Newcastle and Steve Bruce with a win here tonight.
Burnley: Pope, Lowton, Taylor, Cork, Tarkowski, Gundmundsson, Barnes, McNeil, Westwood, Rodriguez, Long
Fulham: Areola, Tete, Andersen, Reid, Loftus-Cheek, Adarabioyo, Lemina, Lookman, Reed, Maja, Aina