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But it needed Sammy Ameobi's spectacular strike and a late Josh Vela effort to progress Phil Parkinson's side to a potentially money-spinning tie.
Ameobi's brilliant 20-yard effort 25 seconds after half-time doubled Wanderers' advantage following Madine's first goal since August 27 in the 44th minute.
Bolton also believed a David Wheater shot had crossed the line - backed up by photographic evidence - with the score at 0-0.
However, the Blades played their part in an entertaining game and Stefan Scougall missed a sitter before Madine's third goal of the campaign.
They deserved to get back in the game through Paul Coutts' deflected effort after 64 minutes, but Vela's superbly crafted goal 20 minutes later appeared to have eased home nerves.
However, Jack O'Connell gave United hope with what eventually proved to be an 86th-minute consolation.
Match report supplied by the Press Association.
Match ends, Bolton Wanderers 3, Sheffield United 2.
Second Half ends, Bolton Wanderers 3, Sheffield United 2.
Foul by Jack O'Connell (Sheffield United).
Jamie Proctor (Bolton Wanderers) wins a free kick in the attacking half.
Jack O'Connell (Sheffield United) wins a free kick in the defensive half.
Foul by Jamie Proctor (Bolton Wanderers).
Goal!  Bolton Wanderers 3, Sheffield United 2. Jack O'Connell (Sheffield United) header from very close range to the centre of the goal. Assisted by John Fleck with a cross following a corner.
Corner,  Sheffield United. Conceded by David Wheater.
Attempt blocked. Leon Clarke (Sheffield United) right footed shot from the right side of the six yard box is blocked. Assisted by Billy Sharp with a cross.
Substitution, Sheffield United. Billy Sharp replaces Stefan Scougall.
Substitution, Bolton Wanderers. Jamie Proctor replaces Gary Madine.
Goal!  Bolton Wanderers 3, Sheffield United 1. Josh Vela (Bolton Wanderers) left footed shot from the centre of the box to the bottom right corner. Assisted by Zach Clough.
Attempt missed. John Fleck (Sheffield United) left footed shot from outside the box is high and wide to the left. Assisted by Caolan Lavery.
Substitution, Bolton Wanderers. Chris Taylor replaces Sammy Ameobi because of an injury.
Attempt saved. Gary Madine (Bolton Wanderers) left footed shot from outside the box is saved in the centre of the goal. Assisted by Zach Clough.
Corner,  Bolton Wanderers. Conceded by Kieron Freeman.
Substitution, Bolton Wanderers. Derik replaces Tom Thorpe.
Chris Basham (Sheffield United) wins a free kick in the defensive half.
Foul by Gary Madine (Bolton Wanderers).
Attempt missed. Kieron Freeman (Sheffield United) header from the centre of the box is too high. Assisted by John Fleck with a cross following a corner.
Corner,  Sheffield United. Conceded by Andrew Taylor.
Attempt missed. Caolan Lavery (Sheffield United) right footed shot from outside the box is too high. Assisted by Leon Clarke.
Caolan Lavery (Sheffield United) wins a free kick in the defensive half.
Foul by Lawrie Wilson (Bolton Wanderers).
Corner,  Bolton Wanderers. Conceded by Aaron Ramsdale.
Attempt saved. Zach Clough (Bolton Wanderers) right footed shot from a difficult angle on the left is saved in the centre of the goal.
Foul by Caolan Lavery (Sheffield United).
Mark Beevers (Bolton Wanderers) wins a free kick in the defensive half.
Kieron Freeman (Sheffield United) wins a free kick on the right wing.
Foul by Josh Vela (Bolton Wanderers).
Mark Duffy (Sheffield United) wins a free kick in the defensive half.
Foul by Gary Madine (Bolton Wanderers).
Corner,  Sheffield United. Conceded by Zach Clough.
Goal!  Bolton Wanderers 2, Sheffield United 1. Paul Coutts (Sheffield United) right footed shot from outside the box to the bottom right corner. Assisted by Caolan Lavery.
Mark Duffy (Sheffield United) wins a free kick in the attacking half.
Foul by Sammy Ameobi (Bolton Wanderers).
Foul by John Fleck (Sheffield United).
Sammy Ameobi (Bolton Wanderers) wins a free kick in the defensive half.
Corner,  Sheffield United. Conceded by Andrew Taylor.
Caolan Lavery (Sheffield United) is shown the yellow card for a bad foul.
Former Sheffield Wednesday striker Gary Madine scored one of the goals as Bolton beat League One rivals Sheffield United to reach round three of the FA Cup.