It was a 20th game unbeaten for the Clarets in the Championship as they stretched their advantage over the Blades to 14 points with just six games remaining.
Johann Berg Gudmundsson, who replaced Jack Cork at the break, scored his first brace for the club after firing two left-footed finishes past Adam Davies.
The visitors had been reduced to 10 men in the 17th minute as goalkeeper Wes Foderingham denied Nathan Tella a clear goal-scoring opportunity.
Depening on United’s result in the early kick-off against Cardiff City at Bramall Lane on Saturday, the table toppers could win the title when they take on Reading at the Select Car Leasing Stadium.
Here are the player ratings as Burnley celebrated boss Vincent Kompany’s 37th birthday with victory on Easter Monday.
1. BURNLEY, ENGLAND - APRIL 10: Johann Berg Gudmundsson of Burnley celebrates with Ashley Barnes after scoring the opening goal during the Sky Bet Championship between Burnley and Sheffield United at Turf Moor on April 10, 2023 in Burnley, England. (Photo by Alex Livesey/Getty Images)
BURNLEY, ENGLAND - APRIL 10: Johann Berg Gudmundsson of Burnley celebrates with Ashley Barnes after scoring the opening goal during the Sky Bet Championship between Burnley and Sheffield United at Turf Moor on April 10, 2023 in Burnley, England. (Photo by Alex Livesey/Getty Images) Photo: Alex Livesey
2. ARIJANET MURIC 7
Didn't have a great deal to do, but made a pivotal stop at the feet of Iliman Ndiaye to prevent Sheffield United drawing level. Showed great awareness to race off his line to quell the danger from Jordan Beyer's back pass. Photo: Naomi Baker
3. CONNOR ROBERTS 7.5
You can't wipe the smile off the right back's face at the moment. Always an option on the outside, pops up in effective positions infield to link play or collect stray balls, and his running off the ball opens up space ahead for his team-mates. Red card worked in his favour, due to Jack Robinson's withdrawal, and looked destined to score with a header from a corner. Photo: CameraSport - Dave Howarth
4. TAYLOR HARWOOD-BELLIS 7
The young Manchester City defender really should have scored when escaping his marker to meet a Johann Berg Gudmundsson corner, but failed to connect with the delivery. Still feeling his way back into things, which is frightening given just how dominant he was against two of the division's best finishers this season. His distribution from the back, as always, was a joy to behold and his delivery for the opening goal was a peach. Photo: CameraSport - Dave Howarth