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United's luck quickly turned however as Ian Moore levelled for the home side and Blake bagged his side's third from the penalty spot on the stroke of half-time.
They pushed for an equaliser early in the second half but the sending off of Michael Brown hampered their chances.
Burnley went in front after 16 minutes when Blake tried his luck from distance. Lee Baxter, the debutant goalkeeper brought in to ease a Bramall Lane injury crisis this week, looked to have the 20-yard effort covered but fumbled the ball into the net.
Baxter's team-mates soon made up for error however as they hauled themselves back level eight minutes later. Michael Tonge had gone close with a shot that was turned away for a corner but Montgomery made no mistake after being fed by Peter Ndlovu on the edge of the area.
Things got even better for the visitors when Whitlow got on the end of a Tonge cross to head them in front in the 31st minute.
The action was relentless and within minutes Stan Ternent's side were back on level terms. This time Blake was the provider as he sent over the cross which Moore managed to meet with a glancing header that looped over Baxter.
The goalscoring was not finished and a dramatic first half ended when Chris Morgan fouled Mark McGregor in the area and Blake tucked home from 12 yards.
The second period did not produce more goals but there was no shortage of incident early on as United started brightly with Tonge, Montgomery and Wayne Allison all taking turns to test Brian Jensen.
Their task was not made any easier however when Brown was dismissed on the hour for serious foul play.
Brown's absence appeared to rob the Blades of any impetus and aside from a few half-chances they could not threaten the Burnley goal.
Tonge and Alan Wright both tried to open up opportunities by getting crosses into the Burnley area but the home side cleared with little alarm and held out for the three points.