French police said one Briton had been seriously hurt in the earlier clashes.
Witnesses said trouble began in the stadium after flares were let off by Russian fans near the end of the game. Some fans had then climbed across barriers designed to keep rival fans apart.
An England supporter, who was in the stadium, told the BBC: "As soon as the Russian goal went in, the Russian guys, sort of, streamed across.
"The segregation wasn`t too great. So it seemed pretty easy for the guys to steam across. It seemed to just sort of separate."
And another fan said: "There were some Russian fans who got same flares in. They were lighting them, there was a loud bang, and then you saw the line of Russian fans... just charging across at the England fans.
"I saw a flare going into the England section or mixed the section. But all England fans just ran across, and as soon as the fans started running everyone ran."
BBC sports editor Dan Roan said England fans began jumping over the perimeter fence in panic.
"Big questions for Uefa here," he tweeted. "Stampede appears to have been triggered by explosion. How did such a device make it inside and where was segregation?"
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`Pretty terrifying`
A senior FA security official said he had not witnessed anything like it in nine major tournaments he had attended.
Ambassador Julian King tweeted: "Number of Brits in hospital overnight. Consular following closely with French authorities."
He added that the embassy was "reviewing the situation" with the French authorities.
Earlier on Saturday French police said one Briton had had to be resuscitated by officers and was left in a critical condition.
They have used tear gas for a third day and a water cannon to try and stop outbursts of violence.
There were also clashes in the city of Nice between police and fans from Northern Ireland and Poland, who are due to play their opening match later. Six people were reportedly injured, before order was restored.
Before the game fans had walked past ranks of police officers in riot gear on their way to the Stade Velodrome venue.
The great majority of fans appeared to be peaceful but there were some reports of trouble.
BBC News correspondent Nick Eardley described the situation as "pretty terrifying" as fans fought with Russia supporters.
He said he had witnessed bloodied England fans being "dragged away by police", to the sound of sirens blaring out across the city`s main square.
Police told the BBC that six people had been arrested and at least five had been injured during the scuffles.
Local police chief Laurent Nunez told AFP news agency: "Police intervened in a clash between English, Russian and French supporters in the Vieux Port district."
Some British journalists in the area have described the scenes on social media, and pointed the finger at a gang of black-clad Russians, who they said had entered the port city`s main square from a side street apparently seeking violence.
The FA has condemned the violence saying it was "very disappointed by the terrible scenes of disorder".
European football`s governing body Uefa said it "firmly condemns the incidents in Marseille", adding that "people engaging in such violent acts have no place in football".
