The visitors had Oli McBurnie sent off on a miserable afternoon, with goals from Jay Rodriguez, Jacob Bruun Larsen, Zeki Amdouni, Luca Koleosho and Josh Brownhill securing a huge win for Vincent Kompany's side.
The victory is just Burnley's second in the league this season, and it has left them in 18th position on seven points, while Sheffield United drop into 19th on five points, with the result putting major pressure on Blades head coach Paul Heckingbottom.
Burnley took the lead almost immediately after the first whistle was blown, with Rodriguez heading a delivery from Charlie Taylor into the back of the net from close range.
Sheffield United had lost 25 of their last 30 Premier League away matches ahead of this contest, and it was not quite the start that the visitors would have had in mind at Turf Moor, with the breakthrough arriving after 15 seconds.
Rodriguez had another opportunity in the 10th minute after smart work from Amdouni, but the experienced forward lifted his effort over the crossbar.
Sheffield United's first half-chance arrived in the 13th minute, with Gustavo Hamer volleying over the crossbar from distance, and Burnley goalkeeper James Trafford then had to keep out a strike from Oliver Norwood, with the visitors starting to offer a threat at Turf Moor.
Burnley managed to double their advantage in the 29th minute, though, with Larsen racing onto a long pass from Dara O'Shea before picking out the bottom corner with an excellent finish.
Cameron Archer had a half-chance for Sheffield United in the period that followed, but he lifted his effort over the crossbar from inside the Burnley penalty box.
McBurnie was perhaps a touch fortunate to escape a straight red card for an elbow on O'Shea late in the first period, and Burnley manager Kompany was particularly unhappy with the decision.
Incredibly, McBurnie did the exact same thing on the same player in the closing stages of the second period, which saw him handed a second yellow card, leaving Sheffield United with a mountain to climb.
Burnley had a chance to score a third in the 54th minute through Koleosho after smart work from the two goalscorers, but the forward fired his low shot wide of the post.
Sheffield United's first chance of the second period came for Will Osula after good play from Hamer, but Burnley goalkeeper Trafford was on hand to make a smart save.
The home side managed to secure the points in the 73rd minute, though, with Amdouni blasting the ball into the bottom corner from inside the box after Sheffield United had only half-cleared a free kick.
It went from bad to worse for the visitors two minutes later, with Koleosho finding the back of the net with a well-placed finish off the inside the post, and Burnley then had a fifth in the 80th minute through Brownhill, who smashed home from the edge of the box after more poor defending from the away side.
Nathan Redmond hit the crossbar late on, with the hosts searching for a sixth, but they ultimately declared at five, with Burnley enjoying comfortably their best game of the season.
Sheffield United now face a home match against Liverpool on Wednesday, while Burnley will be bidding to build on the result when they travel to Wolverhampton Wanderers on Tuesday.No Data Analysis info