The two teams played out a 0-0 draw at Wembley while thousands of members of the Tartan Army descended on central London but were mostly peaceful.
A spokesman said: “The vast majority of these were in central London and in the vicinity of Wembley Stadium.”
Torrential rain had not put off the Scottish fans who took over Leicester Square decked out in their country’s shirts and kilts armed with flags and setting off blue smoke bombs while chanting, singing and playing the bagpipes.
Other supporters gathered at the official fan zone in Trafalgar Square which was set up in a socially-distanced way for 750 key workers, with two large screens either side of Nelson’s Column.
Supporters said they had a “fantastic” time in the fan zone, despite rain throughout most of the match.
Malcolm Gillespie, from Falkirk but living in London, said the draw was an “amazing” result for Scotland and praised the setup at the central London designated key worker area.
The 41-year-old, who attended with his son Santiago, eight, said: “I’m happy but I definitely think we could have edged a win. They’ve played really well.
“It’s been fantastic, a really nice atmosphere.
“I spoke to some English fans who were really friendly. We’ve loved every minute of it.”
On the weather, he said: “It’s OK, we came prepared, we’re Scottish.”
The much-anticipated 115th game in international football’s oldest rivalry is the first time the two sides have met in a major tournament since Euro 96, when Paul Gascoigne’s famous goal sealed a 2-0 win.
Players and staff from both England and Scotland took the knee ahead of the clash.
While the England squad have seen the gesture booed by a minority of supporters of late, Scotland have been standing against racism before games.
Having initially said they would not take the knee, the visitors relented and – despite both sets of supporters booing their rivals’ national anthems – the reaction to the anti-racism gesture was the most positive of recent games, with a cacophony of noise drowning out barely audible booing.
Two men were arrested during early celebrations on Thursday.
A 25-year-old man was arrested for indecent exposure in Hyde Park. He was taken into police custody before being cautioned and released.
Another man was arrested for the attempted theft of a police officer’s hat, assault with intent to avoid arrest and assault of an emergency worker.
He was taken to a south London police station.