Around 3,000 supporters gathered in the Trongate area of the city to celebrate Celtic’s 3-1 victory against Hearts.
Police Scotland said it faced “significant levels of disorder”, which left several officers injured after “missiles, including glass bottles, were thrown at them when assisting with a medical emergency”.
Officers wearing riot gear moved in to clear the area, leaving the streets mostly empty of people by around 7.30pm.
Police said several arrests were made and enquiries were under way to identify others involved, with further arrests expected to follow.
A significant amount of alcohol and pyrotechnics were also proactively seized by officers over the course of the day, the force added.
Assistant Chief Constable Mark Sutherland said: “While we recognise that the majority of fans respected the appeal from police and partners to act responsibly, several thousand still descended on the Trongate area, some of whom engaged in serious disorder.
“Once again, we have seen unacceptable levels of violence and anti-social behaviour and Police Scotland’s officers have been left to deal with the consequences. Being assaulted is not part of the job and regrettably a number of them were injured.
“The levels of criminal behaviour witnessed, in particular the hostility directed towards our officers, is simply unacceptable and cannot be tolerated. Those involved should be in no doubt that they will be identified and arrested.”
Earlier, Hearts said its players and staff were forced to leave the Celtic Park stadium due to the “menacing and threatening atmosphere” created when thousands of fans stormed the pitch after Celtic’s third goal, stopping the game early.
“The pitch invasion caused a chaotic end and nobody seemed to know whether or not the match had been brought to a finish,” Hearts said in a statement.
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The club said it “utterly condemns the shameful scenes at Celtic Park”.
It added: “Reports of serious physical and verbal abuse towards our players and staff, both on the pitch and elsewhere, are deeply disturbing.”
Police Scotland said it was engaging with Heart of Midlothian to establish if any of their players were assaulted on the pitch at the conclusion of the event, adding that the circumstances of the incident were being reviewed.
