A mass exodus from Bramall Lane followed Jay Rodriguez’s 90th minute strike, but those of little faith were suitably punished deep into stoppage time.
Mark Yeates collected the ball on the edge of the area, shifted it onto his left foot and stabbed it past Brian Jensen to send the remaining fans into raptures.
The Blades had fought back from two goals down to get back on level terms before the former Barnsley loanee appeared to have snatched it moments later, but the Blades found what was needed to deny the clarets their first away win of the season.
Gary Speed shuffled his side around once again, Nick Montgomery having to settle for a place on the bench, handing over the captain’s armband to the returning Chris Morgan.
The first half was played at a high tempo, with both sides going close, but neither could make the breakthrough. Yeates’ free-kick beat the wall and the goalkeeper, but couldn’t beat the post, deflecting out for a goal kick.
At the other end Andre Bikey headed over the bar unchallenged, before Chris Iwelumo couldn’t make contact sliding into Chris Eagles’ cross.
Jonny Ertl had earlier forced Jensen to tip over, and Quinn’s low shot kept him on his toes as the first half to an end.
With optimism reigning, spirits were high as the second half begun, but defensive lapses soon brought an end to that.
A cross from the right saw Dean Marney head the ball into an empty goal after Steve Simonsen had rushed off his line and fail to get anywhere close.
Heads dropped, and before any hope of a comeback could be dreamed of, Ritchie De Laet brought down Tyrone Mears in the area to give Chris Eagles a chance to double the visitors lead.
The former Manchester United player placed it into the top corner, and another defeat loomed for Speed’s sinking Blades.
The chances dried up as the Blades probed and their blunt attack lived up to its name, before Mears’ slip under pressure let in Bogdanovic who calmly slotted the ball past Jensen to score his first goal for United in front of the Kop.
The goalscorer was replaced by Ched Evans and then a chorus of groans emanated as Matt Lowton was brought on in place of Jonny Ertl.
The sub payed off just moments later after a teasing cross by man of the match Yeates from the left dropped to Lowton whose side-foot volley bounced, bobbled and struggled before creeping over the line an completing an excellent comeback.
Ten minutes remaining and the game was United’s to win, until a horrendous defensive display allowed Rodriguez to collect the ball on the right stroll into the area and put Burnley back in front.
The fans headed for the turnstiles before Yeates’ late drama in the third of five added minutes rescued a point in a stunning game at Bramall Lane.
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