The Sheffield outfit started the season as promotion favourites, but a miserable run of just one win in nine league games tells its own story.
United, who suffered a 3-0 Carling Cup mauling at the hands of Arsenal's second string in midweek, almost stung Burnley in the sixth minute after Danny Webber found space on the left.
The little striker turned quickly before unleashing a spectacular strike from 20 yards that was tipped around the post by Hungarian keeper Gabor Kiraly.
Burnley, with two ex-Sheffield United players in the side in Andy Gray and David Unsworth, hit back after Robbie Blake and Graham Alexander exchanged passes, Clarke Carlisle beat his marker at the far post only to put his close-range header over the top.
Top-scorer James Beattie latched onto a poor clearance to drill in a 25-yard volley but his shot lacked pace and the Burnley keeper safely smothered the ball.
The ex-Everton striker almost broke the deadlock in the 26th minute but was foiled by a stunning save from Kiraly.
Full-back Phil Bardsley tracked down the left flank and, as Alexander slipped, he pumped over a teasing cross. Beattie produced a stinging header but the keeper conjured up a spectacular one-handed stop.
Kiraly cleared at the feet of Webber as the Sheffield side, who have won just one of their last eight league games, probed for the opener.
Midfielder Wade Elliott should have broken the deadlock minutes after the restart as he connected with a Blake cross but steered his header straight at Ian Bennett, before the United keeper snuffed out Alexander's long-range strike.
Beattie had the ball in the net seconds from time but the goal was disallowed for a foul on substitute Jon Harley.