They passed the ball around sharply, going long at times for variety, the outside centre-backs bombed on as Chris Wilder likes them to, and the centre-forwards linked the play brilliantly. Dean Henderson was able to watch it without having to worry too much about getting his gloves dirty.
The Blades romped into a 3-0 first-half lead which must have left Burnley heads spinning. They never surrendered it.
It was clear right from the off there was going to be no After the Lord Mayor's show from the hosts on their first return to Bramall Lane since beating Arsenal here. To think, a fortnight ago we were worried they were not making the most of their football at home.
It was inside five minutes that Jack O'Connell signalled his intent, getting outside Enda Stevens to win a corner. Chris Basham did the same on the right, only for Matt Lowton's run to block Lys Mousset's path to the cross. In between time, David McGoldrick laid the ball into the path of John Fleck, only for it to get stuck under his feet.
Oliver Norwood showed his hunger to get onto another McGoldrick lay-off which left him second favourite, and when the ball was worked right and back across, McGoldrick missed the target from a tight angle.
A first goal of the season still eludes the centre-forward, but his involvement in the play was excellent.
In the 17th minute, the Blades got their reward. Stevens ball down the line to McGoldrick was exquisite, and when he rolled the ball across, Mousset touched it to John Lundstram, arriving in the box to tap in.
Even the video assistant referee was on United's side, correctly agreeing with referee Simon Hooper that O'Connell had not been able to do anything about a Dwight McNeil cross hitting him on the arm midway through the first half.
It was late in the opening period that Wilder's men effectively won the game.
The Blades were wise enough to keep most of their crosses low with Ben Mee and James Tarkowski waiting to head them away, and left wing-back Stevens's curling delivery in the 43rd minute was perfect. Again Mousset was unable to put it in but Lundstram, arriving at the far post was.
When, a minute later, Fleck latched onto Mousset's wonderful reverse pass and picked his spot, the cheers of celebration had a touch of disbelief to them. You just do not do that to a Sean Dyche side.
All Burnley had offered in the first half was a difficult Ashley Barnes header, running onto a pass from deep, but only able to head the ball up for Henderson to catch. When a chance pinballed to him in the 51st minute, he blazed over.
Inevitably Burnley had more chances in the second half but the relentless pressing the Blades forwards started with and the confidence shown by Baldock chesting a Clarets cross to Fleck deep in his own area and Lundstram in attempting a volley from a tight angle for his hat-trick showed the hosts were undimmed.
McGoldrick headed wide on the stretch after 58 minutes and when he robbed Jack Cork in United's half after 66 minutes, the Blades attacked with the speed and hunger of a team pressing for a last-minute winger. McGoldrick unselfishly played the ball to substitute Billy Sharp, but Nick Pope stuck out a leg to save.
That was a response to Henderson's first save at note, as Tarkowski climbed highest at a corner. Barnes volleyed a Matt Lowton cross into the side netting. There was a late free-kick into the wall by Charlie Taylor.
Lundstram had two further chances for his hat-trick, curling wide after 83 minutes, and dragging a shot in the 90th after being picked out by Sharp.
It was hardly going to spoil his or Sheffield United's day, though. Nothing was.
Sheffield United: Henderson; Basham, Egan (Jagielka 79), O'Connell; Bladock, Lundstram, Norwood, Fleck, Stevens; Mousset (Sharp 65), McGoldrick (McBurnie 75).
Unused substitutes: Freeman, Robinson, Moore, Besic.
Burnley: Pope; Lowton, Tarkowski, Mee, Pieters (Taylor 46); Hendrick, Westwood, Cork, McNeil (Brady 59); Barnes, Rodriguez (Vydra 78).
Unused substitutes: Gibson, Hart, Lennon, Bardsle.
Referee: Simon Hooper (Wiltshire).