At £15 a pop, the half-and-half scarf sellers outside of the Vitality Stadium were cleaning up, and while those souvenirs are often derided as cringe fodder for tourists, there was no blaming those Bournemouth fans queueing to buy one this time.

On the one half, Bournemouth, a club that lost 2-1 at home to Bury in League One 15 years ago on this night. On the other, AC Milan, the Italian behemoths and seven-time conquerors of Europe.

One bloke outside of the Vitality asked for four. He did not even bother haggling. The £60 was worth it so that each of the family had their own red-and-black keepsake.

It was only once inside that those supporters learned their club were gifting them each a free scarf, reading: ‘We are all going on a European tour.’ Not to worry. They would happily take multiple mementoes from this – the night AFC Bournemouth trounced the famous AC Milan.

There was a special feel to it all, from the fireworks exploding over the Vitality to the playing of Andy Williams's The Impossible Dream to the enormous tifo reading: ‘How did we end up here?’

A few in the Milan end may have been asking themselves the same, especially when their once-great team was trailing 1-0 after eight minutes and 2-0 after 32 in this Europa League fixture.

Both goals were penalties – one from Justin Kluivert and the other from Marcus Tavernier – while Milan then had to spend the remainder of this match playing with 10 men as their goalkeeper Mike Maignan was red carded for pushing Tyler Adams in the face.

The Italian journalists in the press box were wearing a told-you-so look at half-time. They had warned us this would happen. ‘You don’t understand,’ one said when asked before kick-off. ‘This isn’t the AC Milan of old.’ Indeed it is not, even if they are unbeaten in Serie A after seven games under Ruben Amorim, the manager formerly of Manchester United.

Nevertheless, this represented a milestone for Bournemouth amid their first-ever season in Europe. They might not have even existed were it not for their former chairman Jeff Mostyn, who was here as a fan, writing another £100,000 cheque to save them from liquidation in 2008.

While Bournemouth were happy to embrace this as an extraordinary match on this occasion, they want these nights to become the norm. Their owner, Bill Foley, is striving for that and victories like this cannot hurt in showing their better players why this club is no stepping stone.

Take Alex Scott as an example. Chelsea want him, and, by all accounts, he wants Chelsea. Foley says they have not given up trying to have him sign a new contract, but like Antoine Semenyo before him, the midfielder is thinking it is time to move to a bigger club. In the words of Foley: ‘Sometimes, players get stars in their eyes and they think Chelsea is the right place to go.’

Bournemouth do not have delusions of grandeur. They know competing with Chelsea, Arsenal, Liverpool and the like will be challenging. But Foley thinks they can become a club where players want to stay rather than leave, and that is where these nights come in.

Bournemouth won here even without Scott, sidelined by a quad injury. The 23-year-old is a wonderful talent, but was not needed. Marco Rose’s side took the lead when Evanilson headed Tavernier’s cross, but Koni De Winter handled the ball on its way wide. Nobody in the stadium noticed it, but VAR Dennis Higler of the Netherlands did, and then referee Vassilis Fotias of Greece did, too, via his pitch-side monitor. Kluivert rolled the ball into the bottom-right corner for 1-0.

Amorim’s anger with his players was already rising in the technical area when Maignan shoved Adams in the face at a Bournemouth corner. The whistle was blown. Again, hardly a soul in the crowd knew what was happening until Maignan was shown a straight red card.

Needing to bring on a replacement goalkeeper in Lorenzo Torriani, the unfortunate one sacrificed was Omari Hutchinson, the 22-year-old whose shin pads carry a picture of himself representing England’s Under 21s. A new taker in Tavernier made it 2-0.

Amine Adli scored a 25-yard free-kick in the 84th minute to cap this victory, with Milan ending the match without a single shot on target. ‘Ruben Amorim, it’s happened again,’ sang Bournemouth’s supporters. Amorim saw the funny side, smiling back at them.

This was a night to cherish for Bournemouth, now winners in all three of their Europa League outings. Indeed, a fair few will hang those scarves in their homes as a reminder. But as a club aiming to go places, they hope this is just the first of many matches with European heavyweights.