Before scoring Flemming had twice been unable to capitalise on opportunities to put the Clarets ahead.
The Dutch forward was unfortunate with a shot that struck Fulham defender Calvin Bassey on the arm but arguably should have done better with a close-range header that was straight at home goalkeeper Bernd Leno.
When he did register his eighth goal of the campaign, it came during a period in which the visitors looked comfortable in the game.
However, Burnley’s defensive fragilities came back to bite them.
The Clarets have now kept just one clean sheet in their past 42 top-flight matches on their travels and on this evidence, it is easy to understand why.
King’s goal arrived directly from a Dubravka error, while Wilson – a player known for his shooting ability on his left-foot – was allowed to drift inside and unload an effort without a challenge being made by three retreating Burnley defenders.
And to compound matters, Laurent’s dismissal for a desperate shove on Jimenez as he faced up Dubravka, was a result of the former Wolves striker being allowed to run through a huge void in the middle of the Burnley back four.