The stroke suffered by left-back Sol Davis on Saturday and the heavy defeats experienced at Everton and Ipswich had left the club looking for a positive result against their Lancashire opponents.
But two first-half goals from Andy Gray put paid to that and Town couldn't even profit from the late sending off of Stephen Foster as they slumped to a fourth blank in front of goal from their last five outings.
Hatters made four changes after their 5-0 drubbing at Ipswich as Ahmet Brkovic, Markus Heikkinen, Richard Langley and Peter Holmes came in for Kevin Foley, Keith Keane, David Bell and Lewis Emanuel.
The lack of defensive cover meant Carlos Edwards and Holmes had to play out of position as makeshift full-backs.
Luton struggled to get a touch of the ball in the opening five minutes as the visitors dominated and yet Town should have gone ahead with their first attack in the sixth minute.
Holmes collected the ball from Dean Morgan and crossed instantly for Brkovic on the edge of the area to touch in behind the Burnley defence for Langley, who somehow fired past the post from eight yards.
As Town settled Morgan created another opening five minutes later as he drove 40 yards through the midfield, played a one-two with Warren Feeney and then curled in a 22-yard effort that Brian Jensen pushed around the post.
Jensen was busy again in the 19th minute when Brkovic prodded on Steve Robinson's corner allowing Rowan Vine to swivel and volley into the goalkeeper's chest from 10 yards.
Hatters' momentum then died down and the Clarets, playing in their all yellow change strip, forced chances with Steve Jones deflecting a James O'Connor shot at Marlon Beresford in the 24th minute before Kyle Lafferty and Jones again struck 25-yard efforts just off target in the space of two minutes.
The openings acted as warnings to Town, but the hosts couldn't have been prepared for what happened next as Burnley scored twice in five minutes.
Firstly Leon Barnett and Lafferty collided as they challenged for a long 31st minute Frank Sinclair pass and, as the ball bounced on, Gray found himself all alone to smartly lob over Beresford from 14 yards.
Gray then got his second as again he was completely unmarked to head home from four yards at the far post after Lafferty found Wade Elliott on the right.
Town ended the half back on the offensive as Barnett headed Morgan's 45th minute corner just wide from 12 yards before Langley sent a 25-yard free-kick over the bar in stoppage time.
The second half began scrappily with neither side able to get the ball down until Luton threatened just before the hour mark as Edwards swung in a deep cross from the right that found Feeney stealing in at the far post, but the Irishman planted his eight-yard header the wrong side of the upright.
With Burnley content to sit back, Town brought on David Bell for Morgan and Adam Boyd for Feeney in an attempt to liven things up.
The move had some effect as, from a half-cleared 70th minute Robinson corner, Holmes' deflected shot fell to Barnett, who lashed over the bar from eight yards.
Three minutes later Town were given hope when Foster was given his marching orders for the Clarets after picking up a second yellow card.
But the best Luton could muster was a 20-yard Bell shot that drifted wide eight minutes from time.