Brentford went ahead in the 85th minute as Ivan Toney headed home a superb cross from Christian Eriksen and matters went from bad to worse for Burnley in stoppage time as Nathan Collins hauled down Toney in the box and was shown a straight red card.
Toney stepped up to convert the spot kick to seal a 2-0 victory which put the Bees up to 30 points, nine points clear of the drop zone, albeit having played two games more than Burnley.
The Clarets stay below the dotted line, Sean Dyche's side remaining third-bottom and two points behind fifth-bottom Leeds but now with only one game in hand.
The Whites would move four points clear of the Turf Moor side if winning Sunday afternoon's clash at home to Norwich City and Burnley play Manchester City next.
Fourth-bottom Everton are sandwiched between Leeds and Burnley but with games in hand on both, the Toffees one point behind United but having played three games less.
Frank Lampard's side are one point ahead of Burnley who have played two games more than them.