In a pulsating final spell, United were denied a winner by a combination of resolute Burnley defending and an adjudged push by James Beattie in the lead up to the goal.
The Blades may have made an inauspicious start to the season but the way they performed belied their lowly league position of 19th.
Bryan Robson made nine changes to the side that lost to Arsenal in the Carling Cup on Wednesday, with only goalkeeper Ian Bennett and Chris Armstrong retained.
Seeking his side's second home win of the season, Robson opted for an attacking trio of Luton Shelton, Beattie and Webber.
The combination looked dangerous from the start and after seven minutes the lively Shelton teed up Webber, who lashed a powerful strike from the edge of the box.
It was tipped around the post by Gabor Kiraly but the chance served to galvanise United, who were then thwarted by Graham Alexander's timely interception of Gary Cahill's cross.
Burnley may have posed an entirely different threat to that of Arsenal, but United on occasions found themselves similarly exposed in defence and were fortunate when Kyle Lafferty fired a good chance over the bar.
That was the pick of Burnley's scoring opportunities in an error-strewn opening half for the Clarets.
Leading scorer Beattie had a header denied by a superb reaction save from Kiraly after Gary Naysmith's surging run and cross and, as half-time approached, Webber came within inches of poking home Shelton's knock-on.
The first blood of the second half went to the Clarets when Robbie Blake chipped a cross into the box but Wade Elliott's header flew straight into the arms of Bennett.
United pressed forward as the gaps began to open up but all too often lacked conviction with their final ball.
At the other end, the Blades' defensive frailties were in evidence again and neat interplay between Andy Gray and substitute Jon Harley allowed Burnley in behind the backline, but Phil Bardsley cut out Harley's resulting cross.
Beattie, searching for his eighth league goal of the campaign, fired a free-kick just over the bar, shortly before Andy Gray shot over from close range under pressure from Naysmith.
In the dying minutes, Armstrong's cross from the left was headed back across goal by Beattie for Webber to nod in but the referee ruled Beattie had pushed a Burnley defender in the process.
The Blades were booed off the pitch by their supporters, despite extending their unbeaten run in the league to four