The defending champions thrilled Celtic Park from the off, Callum McGregor having hit the post and Gary Mackay-Steven passing up a great chance before on-loan Manchester City defender Jason Denayer capitalised on poor set-piece defending to mark his debut with a fourth-minute goal.
The on-going furore surrounding Celtic's reinstatement to the UEFA Champions League at Legia Warsaw's expense continued to dominate the build-up to the match, but a feel-good factor had returned by the time that Kris Commons' free header and Stefan Johansen's strike made it 3-0 before the break.
It meant Dundee United's unbeaten start to the season was in tatters and Celtic were five goals to the good when John Rankin netted a 71st-minute consolation, Anthony Stokes' free-kick and Jo Inge Berget's header extending the glut of home goals.
Berget's second was the final act of an emphatic display and Celtic sit second in the fledging table behind Inverness Caledonian Thistle, who secured a 2-0 win at Motherwell.
Hamilton Academical are up to third, above Dundee United due to the latter's badly hit goal difference, after recording a second Premiership win in four days with a 1-0 success against St Johnstone.
Darian MacKinnon's deflected 15th-minute effort proved to be the only goal of the game, although the midfielder produced heroics at the other end by twice clearing off his own line.
Dundee were awarded a 60th-minute penalty, with the added bonus of Thistle goalkeeper Scott Fox being sent off for his part in a tussle following Danny Seaborne's foul on James McPake.
But substitute goalkeeper Paul Gallacher kept out Gary Harkins' spot-kick and Dundee were unable to make their numerical advantage count.
Josh Magennis and Tope Obadeyi scored headers either side of the break before County's Liam Boyce nodded a goal of his own to set up a grandstand finish.
