Celtic replaced Hearts at the top of the Scottish Premier League for the first time this season after winning a six-goal thriller at Tannadice.
Chris Sutton headed into his own net to give Dundee United the lead but set up John Hartson to fire in the equaliser.
Sutton deflected in a Shaun Maloney drive only for Collin Samuel to hammer home and level for the home side.
But Alan Archibald deflected a Maloney corner into his own net and Stephen Pearson knocked in Celtic's fourth.
United's hopes of preventing Celtic taking over at the top on goal difference had suffered a blow before kick off.
David McCracken and David Fernandez both failing fitness tests, with Garry Kenneth coming into defence and Samuel up front.
But a loose pass back by Bobo Balde presented the home side with a chance after only four minutes.
Goalkeeper Artur Boruc attempted to prevent the ball going for a corner and was lucky to recover in time to block Peter Canero's attempt to slot the loose ball into the net.
Robson's inswinging corner flicked off the head of Sutton and into the Celtic striker's own net.
Celtic were stunned into action and Maloney had claims for a penalty waved away by the referee after a Kenneth challenge by the time the equaliser came after 17 minutes.
Gordon Strachan had given Hartson a starting place, despite Craig Beattie's part in the 5-0 midweek hammering of Motherwell.
And it was the Welsh striker who pounced after Sutton dummied a pass from Paul Telfer to drive the ball under Derek Stillie's despairing hand from 16 yards.
Stillie appeared slow to react but could do nothing when wrong-footed by Sutton's clever toe-poke to divert a long-range effort from Maloney into the net.
United were back level two minutes later when Canero found Samuel on the edge of the box and the Trinidadian turned Stephen McManus and his powerful drive appeared to deceive Boruc.
Another two minutes and another goal, Celtic restoring the lead when a Maloney corner was deflected into the net off the back of Archibald under pressure from Sutton.
Samuel threatened to restore parity early in the second half with a hitch-kick from 10 yards, but the striker's effort curled wide of the far post.
Jim McIntyre did well to fire on target from a difficult angle, but Boruc was well placed at the near post to block for a corner.
Stillie blocked a close-range diving header from Hartson but was left stranded when substitute Pearson pounced on a late ball across the face of his goal.
Dundee Utd: Stillie, Kenneth, Ritchie, Archibald, Wilson, Canero, Brebner, Kerr, Robson, Samuel, McIntyre (Miller 65).
Subs Not Used: Samson, McInnes, Duff, Dodds, Robertson, Cameron.
Goals: Sutton 4 og, Samuel 30.
Celtic: Boruc, Telfer, Balde, McManus, Camara, Maloney (McGeady 90), Lennon, Petrov, Nakamura, Hartson (Beattie 75), Sutton, Beattie (Pearson 83).
Subs Not Used: Virgo, Agathe, Wallace, Marshall.
Booked: McManus.
Goals: Hartson 17, Sutton 28, Archibald 32 og, Pearson 88.
Att: 11,942
Ref: A Freeland