The hosts cruised into a 3-0 half-time lead thanks to goals from Manchester City loanee Jason Denayer, Kris Commons and Stefan Johansen, and twice hit the woodwork in the first half.
Captain Charlie Mulgrew added the fourth just after the break before substitute Jo Inge Berget notched the fifth and sixth, making sure John Rankin’s late strike from distance was a mere consolation and that Deila's side move to the top of the table.
Celtic welcomed former owner Fergus McCann, who saved the club from financial peril in 1994 and steered the club to a position of strength before selling in ’99, on the field before the game.
And he witnessed an explosive start to the game, with Callum McGregor smashing against the post in only the second minute before United burst up the other end with Gary Mackay-Steven, who shot past Craig Gordon but the wrong side of the post.
It took just three more minutes for Celtic to find their opener, however, with Belgian centre-back Denayer pouncing on the loose ball and steering it into the roof of the net on his home debut.
United, who lost several key players over the summer, did threaten again through the lively Mackay-Steven, but Celtic were 2-0 up just before half-an-hour when Commons turned in an Anthony Stokes corner via his cheek.
And yet more poor defending from United, with Paul Paton easily robbed by Johansen on the edge of the box, handed the midfielder space and time to fire past the reach of Radoslaw Cierzniak for the third.
Celtic almost had a fourth before the break when Stokes rattled the crossbar on the stroke of the interval but the goal they wanted came shortly into the second half when Mulgrew flicked in another superb Stokes delivery with the slightest of touches.
And Stokes turned provider again for the fifth – his corner headed back into the danger area by Efe Ambrose and turned into the net by Cardiff City loanee Berget.
The hosts remained hungry for more goals, with Commons and Stokes going close again, but Rankin notched one for the visitors when he exploited a mistake by new boy Denayer to fire home inside the near post on 71 minutes.
Berget was on hand to turn in from close range and bag is second in stoppage time.