Celtic's line-up included Johan Mjallby, in place of Jackie McNamara, and John Hartson was given a rest.
The Celts had an early scare, Billy Dodds had a goal chopped off after six minutes, for a foul by Jim McIntyre on Johan Mjallby.
The disallowed goal hardly spurred Celtic on, the only chances came from Chris Sutton and Henrik Larsson, who proved to be the men of the match.
It was Sutton who grabbed Celtic's first, a slip-up in the Dundee United defence saw Sutton curl the ball home after 33 minutes.
Celtic, in the second half, stepped it up a gear. Rab Douglas replaced Magnus Hedman, who appeared to be struggling with kicking the ball, and Jackie McNamara came on for the yellow carded Stan Varga.
We saw a tremark Henrik Larsson header, at the start of a six minute period, in which the roof caved in on United. Larsson bulleted home a header from a Stan Petrov corner, and the delight on the Swede's face was there for all to see.
He was at it again five minutes later, when a chipped shot rebounded off the crossbar, Chris Sutton was the first player to react and the Bhoys were now cruising.
Billy Dodds was spotted deliberatley going out of his way to blatant bodycheck Larsson. Unbelievably, this assault was missed by three officials and the reptile Dodds was lucky to still be on the park.
Scotland's Goal of the Season competition is now closed - Henrik Larsson has won it. A minute after rattling the Arabs' crossbar, he picked the ball up, went for a dribble and blasted home from the edge of the 18 yard box. The guy is a Living Legend and Celtic must do anything they can to keep him around for at least another year.
Celtic were now 4-0 up and the United fans were seen leaving with only 60 minutes played.
Dundee United grabbed a consolation goal when Didier Agathe was caught in posession. Jim McIntyre struck the ball past Rab Douglas with the aid of a Johan Mjallby deflection.
There was another goal left in Celtic, it came from Chris Sutton. Through into the United box, he was pulled down, and coolly slotted home the resulting penalty. It's Sutton's second hat-trick of the month and well deserved.
The final whistle was the best thing that happened to Dundee United all day, 5-1 to Celtic: that is the way to do it!
Celtic:
Hedman - 7 (Douglas 7)
Balde - 8
Varga - 6 (McNamara - 7)
Mjallby - 8
Lennon - 7
Petrov - 8
Agathe - 7
Thompson - 8
Larsson - 9 (Maloney - 7)
Sutton - 10
Celtic-Mad Man of the Match: Chris Sutton