Postpublished at 21:21 Greenwich Mean Time 18 March 2015
Calum Butcher connects well with a shot from 20 yards after a nice passage of play from Dundee United but the ball flies right into the arms of goalkeeper Craig Gordon.
Jason Denayer heads Celtic in front after 17 minutes.
Leigh Griffiths, Kris Commons and Virgil Van Dijk all score in second half.
Anthony Stokes and Ryan McGowan sent off within minutes of each other.
Scottish Cup quarter-final replay
Winner will face Inverness CT in semi-final
Colin Moffat
Calum Butcher connects well with a shot from 20 yards after a nice passage of play from Dundee United but the ball flies right into the arms of goalkeeper Craig Gordon.
Final switch for the visitors as Aidan Connolly replaces fellow 19-year-old Blair Spittal. A few boos from the home crowd for the newcomer, who won a contentious penalty in the first tie at Tannadice.
Celtic make a change at right-back, with Darnell Fisher taking the place of Efe Ambrose.
Henri Anier is involved in a United attack straight but can't quite find the racing Blair Spittal with his pass into the penalty area.
Henri Anier and Charlie Telfer are on for Dundee United, with Chris Erskine and Ryan Dow making way.
Kris Commons lays off a free-kick for Anthony Stokes to have a crack and Radoslaw Cierzniak has to get down smartly to push the ball away.
BBC Sportsound pundit Willie Miller
"Nadir Ciftci needed to perform tonight for Dundee United to have a chance but he has not reached the standards he is capable of."
United started the second half in sprightly fashion but that Leigh Griffiths strike has knocked the stuffing out of them. Can the visitors regroup, go again and get a goal to make this one interesting again?
Virgil van Dijk strides forward from central defence into the inside left channel as Celtic seek a third goal but United get enough bodies in the way.
BBC Scotland's Tom English at Celtic Park
"That's what the home fans wanted - the assurance goal. Gorgeous delivery from Scott Brown and close-range volley from Leigh Griffiths. Against the run of play, but this thing is done now. I'll eat my hat if it's not. Not that I have a hat."
Not much time for Leigh Griffiths to enjoy that goal as he is withdrawn and John Guidetti will lead the Celtic attack for the remainder of the evening.
Celtic double their lead as Leigh Griffiths watches a long, curling ball from Scott Brown drop over his shoulder and aims a smart volley into the corner of the net.
BBC Scotland's Tom English at Celtic Park
"Dundee United are gaining momentum and Celtic have really lost any sense of urgency. The hosts are asking for trouble if they continue to allow United to set the tempo."
There's another clumsy foul from Efe Ambrose, who has already been cautioned, on Nadir Ciftci. After that reprieve, the Celtic defender heads the resulting free-kick away to safety.
Ryan McGowan powers forward to reach a Chris Erskine pass but the full-back's strong cross finds no takers. Nadir Ciftic then curls in an awkward delivery and Craig Gordon gathers well.
James Forrest skips forward to cause more problems for the United defence only for Leigh Griffiths and Kris Commons to get in the way of each other in the penalty box.
We are up and running again...
Dundee United are still in this game at just 1-0 down but they will need to really up their game after the interval to put the hosts under pressure. Celtic were in cruise control for that first 45 minutes.
The BBC Radio Scotland team are discussing the early booking for Dundee United keeper Radoslaw Cierzniak for clattering into Leigh Griffiths outside the penalty area.
"It was borderline," says Tom English. "It was an assassination!" splutters an incredulous Willie Miller.
There's one other match being played in Scotland tonight and Albion Rovers lead 1-0 at Arbroath at half-time thanks to an early goal from Ali Love.