Mark Duffy bagged a brace while Paul Coutts and substitute Caolan Lavery also scored as the Blades moved second in League One.
The visitors had the better of the opening exchanges but the hosts grew into the game with creative midfielder Duffy at the centre of the action.
Level at half-time, United started the second period with greater urgency and were ahead after 52 minutes.
Duffy's sliced shot was initially saved by Lawrence Vigouroux, but the goalkeeper spilled the ball and was unable to prevent it crossing the line.
The Blades had a second two minutes later as Duffy laid the ball off for Coutts, who smashed home from 16 yards.
Duffy's second and United's third after an hour put the game beyond Swindon, as he slotted past Vigouroux from Billy Sharp's cross.
It was 4-0 when substitute Lavery found the far corner from close range in the 77th minute.
Report supplied by the Press Association.
Match ends, Sheffield United 4, Swindon Town 0.
Second Half ends, Sheffield United 4, Swindon Town 0.
Caolan Lavery (Sheffield United) wins a free kick in the defensive half.
Foul by Raphael Rossi Branco (Swindon Town).
Attempt saved. Caolan Lavery (Sheffield United) left footed shot from the centre of the box is saved in the bottom left corner.
Attempt saved. Leon Clarke (Sheffield United) right footed shot from very close range is saved in the centre of the goal.
Attempt saved. John Fleck (Sheffield United) right footed shot from outside the box is saved in the centre of the goal.
Attempt missed. Nathan Delfouneso (Swindon Town) right footed shot from outside the box is just a bit too high.
Substitution, Sheffield United. Stefan Scougall replaces Mark Duffy.
Caolan Lavery (Sheffield United) wins a free kick in the attacking half.
Raphael Rossi Branco (Swindon Town) is shown the yellow card for a bad foul.
Foul by Raphael Rossi Branco (Swindon Town).
Attempt saved. Caolan Lavery (Sheffield United) right footed shot from the centre of the box is saved in the top centre of the goal.
Corner,  Sheffield United. Conceded by Conor Thomas.
Corner,  Swindon Town. Conceded by Chris Basham.
Corner,  Sheffield United. Conceded by Lloyd Jones.
Attempt blocked. Kieron Freeman (Sheffield United) right footed shot from the centre of the box is blocked.
Substitution, Sheffield United. Leon Clarke replaces Billy Sharp.
Goal!  Sheffield United 4, Swindon Town 0. Caolan Lavery (Sheffield United) right footed shot from the centre of the box to the bottom left corner. Assisted by Ethan Ebanks-Landell.
Corner,  Sheffield United. Conceded by Raphael Rossi Branco.
Attempt saved. Jack O'Connell (Sheffield United) header from the centre of the box is saved in the centre of the goal.
Corner,  Sheffield United. Conceded by Darnell Furlong.
Attempt missed. Caolan Lavery (Sheffield United) right footed shot from the centre of the box is close, but misses to the left.
Caolan Lavery (Sheffield United) wins a free kick in the defensive half.
Foul by Darnell Furlong (Swindon Town).
Chris Basham (Sheffield United) wins a free kick in the attacking half.
Foul by Nathan Delfouneso (Swindon Town).
Attempt missed. Tom Smith (Swindon Town) right footed shot from outside the box is close, but misses to the left.
Foul by Paul Coutts (Sheffield United).
Luke Norris (Swindon Town) wins a free kick in the defensive half.
Substitution, Sheffield United. Caolan Lavery replaces Matt Done.
Substitution, Swindon Town. Tom Smith replaces John Goddard.
Paul Coutts (Sheffield United) is shown the yellow card for a bad foul.
Foul by Paul Coutts (Sheffield United).
Nathan Delfouneso (Swindon Town) wins a free kick in the defensive half.
Corner,  Sheffield United. Conceded by Lloyd Jones.
Attempt saved. Billy Sharp (Sheffield United) left footed shot from the centre of the box is saved in the centre of the goal.
Mark Duffy (Sheffield United) wins a free kick in the defensive half.
Foul by Darnell Furlong (Swindon Town).
Attempt saved. Billy Sharp (Sheffield United) left footed shot from the left side of the box is saved in the bottom left corner.
Sheffield United continued their promotion push with a decisive win over Swindon at Bramall Lane.