Ange Postecoglou’s side looked utterly irresistible in attack as they scored nine goals for the first time in 12 years to inflict United’s worst ever home defeat.
Hat-tricks from Kyogo Furuhashi and Liel Abada blew away their hosts at Tannadice, with Jota, Josip Juranovic and Carl Starfelt also scoring.
It means Celtic are two points clear of Rangers at the summit of the Premiership, while Jack Ross’ side have now lost five on the spin to sit anchored at the bottom with a solitary point from the opening day.
It took Celtic 15 minutes to open the scoring, slicing United apart on the first of many occasions. Abada split the home defence with a through ball, finding Jota, whose cross met an unmarked Kyogo.
Carljohan Eriksson then produced a few fine saves, but his afternoon would only go downhill from there. The second goal was perhaps the pick of the bunch, Kyogo curling in a first-time effort from 25 yards straight into the top corner.
Matt O’Riley combined with Abada to create the third, once again finding Kyogo – who had the freedom of Dundee to tap home his hat-trick.
There was still time in the first half for a fourth. Another tap in, this time for Jota, who may have been offside.
The half-time whistle was sweet relief for United, if only for 15 minutes. Abada got his first soon after the restart when, yet again, Celtic ruthlessly cut their hosts open. Jota found O’Riley who squared across for the winger to add the fifth.
The sixth was comical. Celtic’s free-kick routine resulted in O’Riley drilling the ball into a crouched Reo Hatate, but the ball came back to Josip Juranovic off his team-mate to then drive into the bottom corner from the edge of the box.
Hatate set up the next goal, this time deliberately. Once again, Celtic waltzed through a gaping United defence, with the Japanese midfielder squaring for Abada to finish with ease.
The Israeli scored his hat-trick when he got onto the end of substitute Daizen Maeda’s deflected cross, to dink over Eriksson with his final touch of the game.
Not content with eight, defender Starfelt got in on the act, nodding in a corner to break a record that has stood for 34 years – when they beat Hamilton Accies 8-0 away from home.
Man of the Match – Matt O’Riley
Unstoppable Celtic batter bruised United in record-shattering victory – analysis
Ange Postecoglou said before kick-off that when Celtic are at their best, they are hard to stop. He might have been selling his team short.
Time and time again his side sliced open a porous United defence who looked shell-shocked, conceding eight goals in an hour of football.
Some of the build-up play as Celtic crept towards the box was almost menacing, such was the inevitability of a goal.
They were untouchable, insatiable, irresistible. United had been warned this is what Celtic can do, after Kilmarnock were similarly eviscerated. They did not pay heed.
Twenty-one goals in five games. Their record for a league season is 116.
What they said
Celtic manager Ange Postecoglou: “Our intent is to keep going and not let the scoreboard or the circumstances dictate how we go about things, and the boys were disciplined today.
“We knew Dundee United were going through a rough spell and they were probably a bit vulnerable today. We got on top of them early which allowed us to play our football.”
Both sides are in midweek League Cup action as the competition reaches the last 16. United travel to Livingston, while Celtic travel north to Ross County (both 19:45 BST).