The Cottagers were chasing back-to-back wins for the first time since last July to make up ground in the relegation battle and they went ahead in the 49th minute when Ola Aina unknowingly deflected into the net from Ademola Lookman's corner in the pouring rain.
Up until that point Burnley had been marginally on top in a tight game and Ashley Barnes deservedly brought them level three minutes later as the Clarets maintained their unbeaten 70-year run against Fulham at their home ground.
Fulham's last win at Burnley was in April 1951 but in avoiding defeat here – following a win at Everton last weekend – they inched closer to 17th-placed Newcastle, having trailed the Magpies by 10 points just a few days ago.
Scott Parker's side could have a chance to further reduce the deficit as they take on rock-bottom Sheffield United on Saturday night before Newcastle travel to high-flying Manchester United a day later.
Burnley, meanwhile, stay eight points ahead of Fulham but their injury woes look to show no sign of abating with Johann Berg Gudmundsson seemingly set to join a number of first-teamers on the sidelines while his replacement Robbie Brady lasted just 25 minutes before being withdrawn.
There was very little attacking flair on show in a lifeless first quarter of an hour before Ruben Loftus-Cheek earned an early caution after unnecessarily scything down the returning Charlie Taylor.
Burnley, though, gradually started to impose themselves and Dwight McNeil's inswinging corner was met by Kevin Long, who rose highest to power a header over the bar on his first league start since October 31.
After an innocuous clash with Aina, Gudmundsson – who had scored in Burnley's two previous league games – indicated a problem and then trudged down the tunnel after treatment as he was replaced by Brady.
The hosts ended the half in the ascendancy and the unlikely figure of James Tarkowski sent in a teasing low cross that just evaded the desperate dives of Ashley Barnes and Jay Rodriguez at the near and far post respectively.
Rodriguez was denied by a slight but vital touch by Areola in what was the biggest chance of a largely uneventful first half. However, the game exploded into life four minutes after the interval as Fulham fortuitously went ahead.
Joachim Andersen feathered an attempted backheel volley from Ademola Lookman's corner and the ball ricocheted off the oblivious Aina's chest and into the net, with Brady slicing an attempted clearance at the back post.
Fulham's lead lasted just three minutes though as Rodriguez cleverly skipped into space on the right before teeing up Barnes, whose loss of control wrongfooted Areola and allowed the Burnley striker an easier finish.
Burnley had to make another change shortly after Rodriguez had tested Areola from an acute angle as Brady went off with an apparent hamstring problem to be replaced by Josh Brownhill, making his return from injury.
Josh Maja, Fulham's two-goal hero at Everton, turned and fired low towards the corner but Nick Pope saved well down to his right while Loftus-Cheek was unable to make sufficient contact with Ivan Cavaleiro's corner.
Lookman curled wide on the edge of the area as Fulham were forced to settle for their 10th draw of the season.