Like a song that changes time signature for the hell of it, like a friend that inexplicably blanks you, like a match report that noodles away for ages instead of just telling you what happened, Tottenham Hotspur remain medically incapable of doing things the simple way. This is becoming a kind of mania, a disorder, a cry for help. What is this? Who are you really? And, you know, can you not?

For all this, Ange Postecoglou’s side are League Cup semi-finalists, the latest plot twist in a season where nobody can really agree whether things are going well or not. Great football. But also some terrible football. But also, three games from a trophy. But also, 10th in the Premier League. But also two goals for Dominic Solanke. But also two goals basically given away by Fraser Forster.

Related: Tottenham Hotspur v Manchester United: Carabao Cup quarter-final – live

And this game was the video explainer you would give to someone who had never seen Tottenham play before. Three goals in a rampant first hour: Dejan Kulusevski and James Maddison running amok, Yves Bissouma a pillar of poise in midfield, Djed Spence and Archie Gray sparkling in unfamiliar defensive roles. Then the childlike defensive implosion to allow Manchester United a richly undeserved path back into the game. Then Son Heung-min scoring directly from a corner. Then, finally, another United goal to make it 4-3 because, you know, why not?

There will be a temptation for United to take the positives out of this. To put the Spurs goals down to freakish individual errors, to put their own down to a voracious pressing game being further honed under Ruben Amorim. This is an urge they probably need to resist. Amorim is a smart enough manager to harbour no illusions about the scale of the shambles still awaiting him, a squad that basically looks three pints to the wind and in need of a little sobering up.

For all the vague Jesus vibes that have pursued Amorim since joining the club, perhaps this was a valuable reminder that these are many of the same players who thrashed and flailed so ineffectively under Erik ten Hag, a combination of the once good enough, the potentially good enough and the not-quite good enough. Antony and Victor Lindelöf and Jonny Evans are still hanging around the place. Altay Bayindir has three years left on his contract and definitely should not be allowed all of them.

There is still a new system being seeded and sowed. Marcus Rashford is still marooned on the thinking chair. And for all their new energy and thrust, United still look uncomfortable in defence, disturbed by teams who make them turn and run. One clean sheet in eight games tells its own story. For an hour Spurs simply ripped them up, a version of themselves that Postecoglou must wish he could roll up in a duffle bag and take everywhere with him.

This is Spurs when it all makes sense: full of hard running and clever angles and flicks upon flicks. When the players are largely interchangeable because the parts are meant to interchange anyway. Spence, a right-back at left-back. Gray, a midfielder at centre-half. Kulusevski on the right but occasionally moonlighting on the left. Passages of humming chaos in which possession is lost, regained, lost again, regained again, to the point where you’re not quite sure whether they’re attacking or defending.

These were the combinations that produced the first goal, Pedro Porro with the shot from distance, Bayindir parrying the ball into the path of Solanke, who buried the rebound first time. United had spells of possession but the final pass was often lacking, and with a diet of long balls being pumped up to Rasmus Højlund the wider strategy was not entirely legible.

It was still not legible when Spurs doubled their lead 47 seconds into the second half. As Son drove through the centre, as Maddison overlapped on the left, United were still grasping at phantoms, herding and narrowing, a safety in numbers that was really no safety at all. Kulusevski slammed the ball in from close range after a non-clearance by Lisandro Martínez.

Spurs get Liverpool in semi-final draw

Tottenham's reward for beating Manchester United was a semi-final with Liverpool in the Carabao Cup. Spurs will host the Premier League leaders first with the return leg at Anfield. It opens the door to a north London derby in the final with Arsenal taking on Newcastle in the other semi-final. The Gunners will host Newcastle before the second leg at St James' Park. The first legs are on the week of 6 January with the second legs starting 3 February.

When Solanke created and fired home the third emphatically in the 54th minute, it was game over, at least unless Spurs did something unutterably stupid. Like giving the ball away to Bruno Fernandes five yards from goal. Or letting Amad Diallo tackle the ball into the net. Well, you won’t believe what happened next!

First Forster and Radu Dragusin shared an awkward moment, Fernandes stole in and substitute Joshua Zirkzee had a tap-in into an empty net from two yards: the sort of range from which Zirkzee, and indeed your most elderly relative, is deadly. Next Forster dithered over a clearance, Diallo put in a speculative slide and Forster – a man older than many countries – obligingly smacked the ball straight at him.

There were a couple more late scares, even after Son scored despite United howling for a foul on Bayindir. Evans headed in to stir a little undeserved jeopardy into the final seconds. But Spurs held on, as they hold on to the dream of a first trophy since 2008. It would be a profoundly strange thing to happen. But Spurs are turning into a profoundly strange team.