Jason Denayer, the 19-year-old Belgian defender who arrived on a year-long loan from Manchester City last week, took only four minutes to make his mark by lashing the ball in from close range before goals by Kris Commons and Stefan Johansen had the points tied up by the interval.
In manager Ronny Deila's first game at Celtic Park since taking over from Neil Lennon, stand-in skipper Charlie Mulgrew claimed a fourth and Jo Inge Berget notched a fifth- his first since arriving on loan from Cardiff City - before Dundee United captain John Rankin pulled a goal back with Berget able to make it six in the last seconds.
While United massively contributed to their own downfall, Celtic's convincing victory was perhaps apt on the day that Fergus McCann made a rare return to Parkhead.
To huge cheers from the Hoops fans the 2013-14 Premiership flag was unfurled by the wealthy businessman who had saved the club from bankruptcy in 1994 before implementing plans for the modern-day Celtic including the refurbishment of the stadium.
It was an opportunity to atone for those Hoops fans who had booed McCann, an expatriate Scot living in Canada, when he performed the same duties 16 years ago, some supporters unhappy that he refused to spend big money at the time due to a financial foresight that has long since been vindicated.
McCann had barely wandered up to his corporate seat in the North Stand when Parkhead midfielder Callum McGregor slammed a shot against the post.
The visitors missed a great chance seconds later when Gary Mackay-Steven raced on to a Stuart Armstrong pass and slipped the ball keeper Craig Gordon but also the far post.
However, when United defender Callum Morris failed miserably to deal with Commons' corner from the right, the unmarked Belgian slammed the ball high into the net from eight yards.
There was some slackness in the Hoops defence, perhaps understandably with Denayer starting for the first time and Efe Ambrose a late change after Adam Matthews had to withdraw with injury before kick-off.
United looked purposeful moving forward but much more susceptible in defence than the champions and that showed in the 27th minute when Commons all too easily nipped in at the near post to head in Anthony Stokes' corner from the left.
There was more unnecessary woe for the Tannadice men in the 33rd minute when midfielder Paul Paton was easily dispossessed by Johansen who ran through to drive low past Radoslaw Cierzniak.
As rampant as Celtic were, they were still gifting the Taysiders chances with Gordon having to rush from his goal to thwart Nadir Ciftci.
Mario Bilate replaced the injured Stuart Armstrong before the interval and he watched as Berget rattled a shot against the bar before Stokes' mis-hit shot tipped over the bar by Cierzniak in the last action of the half.
Beram Kayal replaced Johansen for the start of the second half but there was little apparent disruption as Cierzniak soon found himself having to make a good save from Stokes' angled-drive.
But Celtic also remained slack in defence with Gordon forced into a fingertip save from Ciftci's low drive at the expense of a corner which came to nothing.
However, United had more calamities to reveal and in the 54th minute Stokes' free-kick from wide on the left was missed by a number of Tannadice players with Mulgrew claiming to have got a nick on it as the ball ended up in at the far post.
Cierzniak remained under siege and he could not prevent Berget getting his first Celtic goal in the 62nd minute as Ambrose headed a Stokes corner back cross goal, allowing the Norwegian to bullet a header in from a few yards out.
Rankin's deflected drive from the edge of the box which nestled behind Gordon did little to stop Celtic pushing for more and in the closing stages Berget grabbed a second on the break to complete the rout.