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An estimated crowd of 70,000 crammed into the Portuguese National Stadium in Lisbon to witness the Glasgow side lift the greatest prize in club football.
Milan have been champions of Europe three times in the past four years and this was only their second defeat in continental competition in that time.
As the final whistle blew, euphoric Celtic fans poured onto the pitch to celebrate their team's victory, many whooping with joy and waving banners.
The manager, Jock Stein, said: "There is not a prouder man on God's Earth than me at this moment. Winning was important, but it was the way that we won that has filled me with satisfaction.
"We did it by playing football; pure, beautiful, inventive football. There was not a negative thought in our heads."
According to the Celtic players, Stein told his players to "go out and enjoy themselves" at the start of the match.
But it could all have turned out very differently. Within minutes of kick-off defender Jim Craig felled Renato Cappellini and Alessandro Mazolla netted the resulting penalty.
Milan held onto their early lead until half-time. But shortly after the break Celtic full-back Tommy Gemmel scored the equaliser
The goal gave Celtic the inspiration the players needed. They continued to attack the Italian goal until Gemmel again stormed up the left wing, passed back to Bobby Murdoch who sent a powerful shot towards the goal which was deflected into the net by Stevie Chalmers to give the Glasgow side a 2-1 lead.
The celebrations began immediately and although the Portuguese police feared the crowd would get out of control, there was no hooliganism.
But the chaos inside the stadium meant that the Celtic players could not be presented with the trophy on the pitch.
Instead club captain Billy McNeill had to be ushered round the outside of the stadium under armed escort. He then climbed the stairs to the presentation podium where he finally held the trophy aloft to enormous cheers from the crowd.
Jubilant fans danced in the streets of Glasgow after hearing of their club's 2-1 win .
The team is expected to fly into Abbotsinch Airport in Glasgow tomorrow night, from where they will drive to Celtic Park for a heros' welcome from fans.
The Celtic victory is regarded as the greatest in the Scottish club's history.
The 11 players became known as the Lisbon Lions - the first non-Latin side to win the European championships.
The team players were all born within a 30-mile radius of Glasgow.
They flew into Glasgow the following day and were transported by coach to Celtic Park where an estimated 50,000 people had packed into the grandstand and terraces to greet their heros.
Jock Stein was credited with creating the winning side. He had joined as manager in March 1965 and within weeks Celtic had won the Scottish Cup for the first time in 11 years - they had previously been beaten in four finals.
He led the team to many more triumphs - including a second European cup final four years later, but the team lost to Dutch side Feyenoord Rotterdam.
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