The visitors came flying out the blocks, and following some sloppy early defending by the Clarets, the Cottagers found themselves two goals up inside the first 25 minutes of the match, with Joao Palhinha and Rodrigo Muniz both finding the net.
Burnley put on a better showing in the second period and managed to clinch what could be a vital point at the end of the season with Chelsea loanee Fofana scoring twice after coming off the bench.
Fulham secured a deadline day loan deal for Chelsea forward Armando Broja, who started on the bench with Marco Silva opting to maintain faith in Brazilian forward Muniz from the off.
In one of two changes to the side that faced Manchester City in midweek, Vincent Kompany handed a league debut to one of his January loan signings, Lorenz Assignon, in place of Ameen Al-Dakhil. Aaron Ramsey was also brought into the starting 11 at the expense of Johann Berg Gudmundsson.
Fulham, who were winless in their last 10 Premier League away games, including five defeats in a row and failing to score the last three, brought an end to their goal drought on the road as Palhinha rose above Wilson Odobert to head home a near-post corner from Andreas Pereira inside the opening 20 minutes of the match.
The Cottagers had not won a single of their last 28 matches at Turf Moor (W22 D6) since April 1951 but took a huge step toward ending that run just three minutes after the opening goal when Muniz beat the offside trap to get on the end of a flick-on from Antonee Robinson and lobbed James Trafford, who could not keep out the shot despite getting a glove it.
That marked the Brazilian's first goal for The Cottagers in his 11th appearance and could not have come at a better time for him, with the newly-arrived Albanian now breathing down his neck for a spot in the starting lineup.
Fulham were purring as the half-hour mark approached, and they almost found a third first-half goal as Pereira got in behind The Clarets defence from a diagonal ball from Robinson and raced into the opposing box before firing a near-post shot which Trafford palmed away for a corner.
As the half wound down, Burnley started seeing plenty of the ball but were doing very little with it, and even when they did win the ball back high up the pitch, they lacked the quality to find incisive passes to unlock the Fulham defence and create clear-cut chances.
Boos sounded out around Turf Moor as the sides headed down the tunnel at the break, though Kompany's halftime speech seemed to have an instant impact as The Clarets came out with more intent in the second period and had a couple of presentable chances inside the first two minutes of the restart but both Zeki Amdouni and Josh Brownhill had their shots at goal blocked.
After weathering the early second-half onslaught, Fulham began dictating play as they approached the hour mark and almost put the match to bed when the Brazilian duo of Pereira and Willian worked a neat one-two from a short corner with the latter firing a shot at goal at the end of the move which forced a very good low save from the Burnley keeper.
Kompany opted to bring on Fofana in the final 20 minutes, and the Ivorian made an instant impact as he got on the end of an inch-perfect cross by fellow January arrival Assignon and nodded home to give Burnley some hope of a comeback just moments after stepping on to the pitch.
The hosts piled on the pressure as the match entered stoppage time and they got their reward in the 92nd minute when Odobert charged past Tete down the left flank and crossed to Fofana, who bundled home his second of the afternoon, and Burnley's equaliser.
With that, Fulham move up to 12th place on 26 points while Burnley remain second bottom on 13 points and seven points from saftey.
Both sides will return to league action next Saturday, when the Clarets head to Anfield to face Liverpool, while Fulham return to Craven Cottage to face Bournemouth.