The win strengthens the Blades' grip on a play-off place and if they continue to perform to a similar standard until the end of the season, automatic promotion is still not out of the question.
Brown and striker Steve Kabba, who scored the winner against Leeds on Sunday, were magnificent and the margin of victory could have been greater although Gareth Taylor and Robbie Blake did get on the scoresheet for the visitors.
Defeat would appear to extinguish Burnley's play-off hopes but they should have gone ahead after 13 minutes when Lee Briscoe was played in by Glen Little but his right-foot shot from 15 yards was too close to Paddy Kenny who made a comfortable save.
It proved a costly miss as midfielder Brown fired Neil Warnock's side ahead with an excellent finish three minutes later.
Phil Jagielka squared the ball to the 26-year-old and he took advantage of oceans of space to curl a right-foot shot into the top right-hand corner from 22 yards.
Paul Weller was booked for dissent at the midway point of the first half before the lively Kabba, after three goals in his last three games, pounced on a mistake by Ian Cox but Marlon Beresford plunged at his feet to prevent him doubling the lead.
Tackles were flying in with the recalled Little next to go in the referee's notebook, but it was United who continued to impress and they doubled the lead 10 minutes before the interval.
Again it was Brown who found the target, firing a free-kick from the left-hand side of the area over the wall and into the corner after Kabba had been fouled by Little.
The home side were rampant now with Burnley's ageing rearguard all over the place and Dean Windass wasted a golden opportunity to make it three before Beresford saved efforts from Nick Montgomery and Kabba.
Stuart McCall was forced off with Peter Ndlovu his replacement just before the break but with Brown continuing to dictate affairs and Kabba a constant menace, the second period began in the same vein as the first had ended.
Shaun Murphy received a yellow card for a crude challenge on Ian Moore before former Bramall Lane striker Taylor gave the visitors hope when his downward header following Briscoe's left-wing cross reduced the arrears after 63 minutes.
But the two-goal advantage was restored three minutes later when Kabba chased a lost cause and pulled the ball back from the right byline for substitute Peter Ndlovu to arrive at the back post and steer the ball home from 12 yards.
The Zimbabwe international scored his ninth goal of the campaign after 71 minutes, cutting in from the left after Montgomery's cross-field ball to drive a right-foot shot past Beresford from 20 yards.
Ndlovu had chances to complete his hat-trick but Burnley substitute Blake pulled one back with a minute remaining when his angled left-foot shot defeated Kenny.