There was a heavy police presence in the area as fans arrived to enjoy a pre-match drink ahead of the game, but the atmosphere turned ugly about an hour before the 4.05pm kick off, leading police to close the road briefly.
Police said they had detailed 13 people as a result of the incident.
Witnesses reported seeing “running battles” between the police and Rangers supporters, while pictures appeared to show dozens of officers surrounding, or “kettling” a section of the Rangers support at the West End of the Grassmarket.
One witness said the fighting had spilled out of a Grassmarket bar and said bottles had been thrown by supporters.
One eye witness, who had been in the Black Bull pub when the incident broke out, said: “There was a flood of people running outside, there were about four or five police vans.
“It was busy with football fans, who started coming through the Grassmarket at about 12. They were running and chanting.
“I think there were about 60 of them in the pub beforehand.”
And @mainbart tweeted: “Terrible scenes at #Grassmarket absolutely shocking horrendous stuff #Riot.”
A Police Scotland spokesman said: “There was a disturbance prior to the match in the Grassmarket between two groups. 13 people were detained at taken to St Leonard’s police station.”
The disturbance broke out just after 3pm, and the vans arrived on the scene shortly afterwards, having already been deployed to cover the Cup Final.