Kyogo Furuhashi and Liel Abada both scored hat-tricks, while Jota, Josip Juranovic and Carl Starfelt were also on target as the Premiership champions recorded their biggest win since defeating Mark McGhee's Aberdeen by the same score in 2010.
Celtic cut United open time and again to leave Tannadice manager Jack Ross facing major questions with his side now conceding 23 goals in just four matches.
The victory puts Celtic back on top after their fifth win from five games.
Celtic manager Ange Postecoglou made three changes. Starfelt and Abada were handed their first starts of the season while Reo Hatate was recalled. Giorgos Giakoumakis was absent.
Ross made four changes as he tried to halt a four-match losing streak. Tony Watt and Charlie Mulgrew were among those dropping out as Ross brought in Ross Graham to join a back three with Archie Meekison handed his first start of the season in midfield.
United actually started well and Glenn Middleton had a shot on the turn pushed away by Joe Hart, who beat Steven Fletcher to the rebound at the expense of an accidental boot to the head. Fletcher was belatedly booked by Steven McLean.
Hart was able to continue after lengthy treatment and a change of jersey and shorts, and Celtic soon took the lead in the 15th minute.
Abada sent Jota in behind and Furuhashi had kept himself ahead of the United defence to allow himself time to take a touch and slot home.
Carljohan Eriksson kept his side in the game with saves from Matt O'Riley, Hatate and Abada, and Middleton fired just wide from 20 yards after Scott McMann won the ball deep in the Celtic half.
The flurry of goals started in the 40th minute when Furuhashi swept the ball first time into the top corner from 22 yards after O'Riley robbed McMann.
Celtic went for the kill and scored a brilliant third in stoppage time. A first-time pass from Juranovic, a dummy from Hatate, and a first-time cross from O'Riley opened United up and Abada squared for Furuhashi to tap in. There was a suspicion of offside but it was too close to call.
The visiting fans were singing "piling on the agony, putting on the style", a song about their 7-1 League Cup final win over Rangers, and it was clear their team were intent on following suit.
Celtic showed the urgency of a side chasing a last-minute equaliser as they searched for a fourth before the break and they duly got it. O'Riley's touch sent Abada in behind and he crossed for Jota to net from close range. The Portuguese winger was several yards offside but the goal stood.
Half-time brought respite but no shift in momentum. Callum McGregor's pass to Jota put United on the back foot and O'Riley squared for Abada to knock home in the 50th minute.
Juranovic drilled home five minutes later after O'Riley's free-kick had hit the wall and, after Fletcher hit the post at the other end, number seven came just before the hour mark. Jota found Hatate's run and the Japanese midfielder squared for Abada to knock home.
Postecoglou soon brought on four substitutes and there was a temporary lull in proceedings before Abada dinked the ball home after Daizen Maeda had helped on Aaron Mooy's lofted ball in the 77th minute.
Starfelt headed David Turnbull's corner in off the post four minutes later and the Celtic fans chanted: "We want 10".
They came close on several occasions. Turnbull's header appeared to bounce over the line and spin back, Starfelt had a header cleared off the line, McGregor swept a shot just wide, but United were spared the further embarrassment of conceding double figures.