United were playing under new boss Micky Adams for the first time and he watched on in delight as Mark Yeates fired the visitors ahead in the ninth minute.
Winger Chris Eagles brought the Clarets level in the 29th minute and two goals on the stroke of half-time from Chris Iwelumo and Jay Rodriguez put the hosts 3-1 ahead at the break.
A penalty from the in-form Ched Evans shortly after the restart hauled the visitors back into contention but Burnley added a fourth late on to seal victory through substitute Steven Thompson.
Stuart Gray was placed in caretaker charge of Burnley following the departure of Laws earlier this week and he opted to start with Eagles, Iwelumo and Rodriguez as his side's attacking trio.
Chances came and went right from the outset and midfielder Dean Marney threatened inside the first minute when he saw a shot blocked by Nyron Nosworthy.
Rodriguez then headed wide moments later before the Blades went close themselves as Evans fired goalwards from 12 yards to force a fine parried save from Lee Grant.
However, Evans soon fired home right-footed from 12 yards out to put the visitors ahead and Burnley quickly sought an equaliser.
Veteran midfielder Graham Alexander had a shot blocked by Lee Williamson and then Iwelumo saw a header saved by Richard Wright.
Burnley finally drew level in the 29th minute when Eagles drilled a superb shot into the net from 30 yards.
The Clarets began to assume control as the interval approached and scored two goals in quick succession.
In the 45th minute, Iwelumo headed into the bottom corner from 12 yards and then Rodriguez swept home a clinical right-footed shot from 12 yards out.
The Blades replaced Wright with Steve Simonsen between the posts at half-time but a penalty soon dragged them back into contention within two minutes of the restart.
Wade Elliott fouled Leon Britton and Evans fired home from 12 yards out, giving the Blades hope of a comeback.
Jamie Ward tried his luck from distance but Burnley went close to grabbing a fourth as Iwelumo fired over the bar from just six yards out.
Caretaker boss Gray then replaced Iwelumo with Thompson as they sought to close the game out.
The moved proved a masterstroke as Eagles showed his quality with a fine pass to Thompson, who showed composure to rifle home right-footed from six yards out.