The 20-year-old striker - in for the injured Leigh Griffiths - started the rout when he headed the home side ahead from a Scott Sinclair corner in the 33rd minute.
The Frenchman added a second with a confident finish before Gers striker Joe Garner reduced the deficit just before the interval.
However, hopes of a comeback were dampened when Sinclair made it 3-1 before Rangers debutant defender Philippe Senderos was sent off by referee Willie Collum for picking up two bookings, the second for a hand ball.
Dembele added a fourth and substitute Stuart Armstrong a late fifth as the visitors ended the match in disarray, with the Parkhead side going four points clear of their newly-promoted rivals at the top of the table with a game in hand.
Even this early in the season - the Hoops have only played four league games - it looks good for Brendan Rodgers' side in their quest for title number six and they now prepare for their Champions League opener against Barcelona at the Nou Camp on Tuesday night.
There was a pre-match blow to Hoops hopes when star striker Griffiths failed to recover from a hamstring injury - he is doubtful for the game against Barcelona on Tuesday - with French Dembele coming in but the home fans had no need to worry, as it transpired.
Former Arsenal defender Senderos made his debut in the heart of the Gers defence and Garner made his first start in an attack-minded line-up which included Niko Kranjcar, Josh Windass and Barrie McKay.
As always in this fixture, the atmosphere was super-charged and the frenetic early stages saw Rangers cope comfortably with Hoops' attacks while looking purposeful moving forward.
However, for all Rangers looked mostly untroubled they were found wanting again at a set-piece when Dembele beat Gers defender Rob Kiernan to a Sinclair corner to head in off the post.
The Govan side wobbled with the set-back and when Kiernan gave possession away, midfielder Nir Bitton sent Dembele racing through a massive gap in the Gers rearguard and he turned inside Senderos before knocking it past Wes Foderingham, who not for the first time, found himself unguarded.
Barely had the cheers from the delighted Celtic fans subsided when Rangers right-back James Tavernier crossed to the back post for ex-Celt Kenny Miller to head back across goal, giving Garner the chance to head in from under the bar.
Midfielder Andy Halliday replaced Kranjcar for the start of the second half and it was Rangers who came close to levelling when Barrie McKay curled a shot past the far post.
Armstrong came on for Tom Rogic in the 54th minute and six minutes later the Parkhead side restored their two-goal lead when Dembele slipped in Sinclair who beat Foderingham at his near post.
More substitution followed, Patrick Roberts for Forrest, Martyn Waghorn for Miller and Harry Forrester for Kiernan but Celtic had a firm grip on the game.
Foderingham made a fine save from Armstrong's long-range effort before Callum McGregor replaced Hoops skipper Scott Brown, whose much anticipated battle with Joey Barton did not materialise.
When Senderos was sent off soon afterwards - he had been earlier booked for a foul on Sinclair - it was effectively over for Rangers, who then had assistant Davie Weir sent to the stand.
Celtic kept pressing and Dembele grabbed his third when he had a time and space to take down a Mikael Lustig cross and fired into the far corner, with Armstrong adding more salt to some very blue wounds when he made it 5-1 in added time.
TWEET OF THE MATCH - "Easy, easy, easy,.... #OldFirm @celticfc," Tenerife's former Celtic midfielder Marc Crosas (@marccrosas).
STAR MAN - Moussa Dembele has had a frustrating time since swapping Craven Cottage for Celtic Park, with his own patchy form and the red-hot instincts of Griffiths keeping out of Rodgers' team. But with the Scotland striker ruled out injured, the 20-year-old was given the opportunity to spearhead the Hoops attack and made the most with three clinical finished and another assist to boot.
DATA POINT - Old Firm hat-tricks are rarer events than finding hen's teeth. Demeble's trio of goals was the first in the fixture since 1984, when Ally McCoist bagged the match ball as Rangers claimed the League Cup with a 3-2 win in the Hampden final. The last Celtic player to hit the net three times in a derby clash with the Ibrox men was Stevie Chalmers back in January 1966.
REF WATCH - Tensions were always going to be running high as Rangers made their first visit to Glasgow's east end in four years. But Willie Collum will be grateful that the players did not get too carried away amid the feisty atmosphere. The official still had to dish out six bookings, a red card and send Rangers assistant boss Davie Weir to the stand but it was a controlled display from Scotland's top referee.
Celtic - Dorus de Vries: 7 (Out of 10)
Mikael Lustig: 7
Kolo Toure: 7
Erik Sviatchenko: 6
Kieran Tierney: 6
Nir Bitton: 6
Scott Brown: 6
James Forrest: 6
Tom Rogic: 6
Scott Sinclair: 8
Moussa Dembele: 9
Stuart Armstrong (on 53, for Rogic): 6
Patrick Roberts (on 64, for Forrest): 5
Callum McGregor (on 74, for Brown) 5
Rangers - Wes Foderingham: 5
James Tavernier: 5
Rob Kiernan: 5
Philippe Senderos: 4
Lee Wallace: 5
Joey Barton: 5
Niko Kranjcar: 4
Josh Windass: 7
Kenny Miller: 6
Joe Garner: 6
Barrie McKay: 5
Andy Halliday (on 46, for Kranjcar): 5
Martyn Waghorn (on 63, for Miller): 4
Harry Forrester (on 71, for Kiernan): 4
WHO'S UP NEXT?
Barcelona v Celtic (Champions League, September 13)
Rangers v Ross County (Scottish Premiership, September 17)
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