Hennessey was called upon to push away Barnes' curling strike to ensure Palace remained just one goal behind, but that was a rare attack from Burnley in a half during which the visitors saw much of the ball.
Lee Chung-yong's blunder today allowed Burnley to take advantage of the chance that was presented to them and put Crystal Palace firmly in the history books.
This week, amidst newspaper reports that his Crystal Palace career might be shorter still, he met with the chairman Steve Parish. "We are four games in, it's a bad start but we have to stick together".
Palace have made a stuttering start to the season and are still searching for their first goal of the new Premier League season under their new manager. Perhaps he will be afforded a reprieve. As of Monday he will be up to 77 at Palace.
De Boer's men kept the Clarets pinned back but couldn't find a way through to seriously test Pope as his team-mates defended with great discipline to protect their lead. Sadly for the recalled Lee, the defence he bisected was his own.
Record Clarets signing Chris Wood is looking to earn a better reputation than the arsenal of strikers that have preceded him at Turf Moor.
Crystal Palace pay a visit to Burnley on Sunday in Matchweek 4 of the 2017/18 Premier League campaign. "I told (De Boer) so". Or, in this case, not to.
De Boer will be encouraged by the effort and commitment of his team, who will leave Lancashire feeling they deserved at least a draw, but ruing their failure to convert their overwhelming possession after the break into a goal. Their methodology - set-pieces and an aerial attack - may have owed more to Sam Allardyce than his successor, but it nearly succeeded. A Dann shot from a corner was cleared off the Burnley line by Matt Lowton and then Benteke headed just wide from an Andros Townsend cross.
However, three defeats from three Premier League matches have left the Dutchman's future under scrutiny. Van Gaal offered his protégé tips for succeeding in England based on his two years at Manchester United, from which he apparently carries a curious disappointment that De Boer is determined to avoid. But De Boer certainly hasn't had any luck on his side.
Yet at home they have yet to score a goal or earn a point. They have to recognise "Okay, now we have to do this, now we have to do that".