A second group of protesters blocked the entrance to the Estonian embassy, preventing Marina Kaljurand, the ambassador, from leaving for the news conference venue on Wednesday afternoon.
In Brussels, the European Commission expressed concern over the protests at the embassy and said the EU would formally raise the matter with Russia.
In Tallinn,the Estonian capital, the foreign ministry said it had evacuated the family of diplomats and protested strongly.
'Physical attacks'
Estonia responded to the attacks by telling Russia "to remain civilised".
"Physical attacks on the Estonian Republic's diplomatic representatives are unprecedented and absolutely unjustifiable," a statement issued by the Estonian foreign ministry said
"These attacks call for immediate actions from the international community."
About 25 demonstrators shouting "shame on Estonia" and "fascism will not be allowed" burst into the press centre at a Russian newspaper in central Moscow minutes before the planned midday (0800 GMT) start of the news conference.
The ambassador's bodyguards used crowd control gas to clear the hall. Estonia said its ambassador's car was also attacked and its flag torn off.
Maxim Mishchenko, a leader from the Young Russia group, said: "We can't allow her [Estonia's ambassador] to be here until her government and her president, who has called our grandfathers gangsters, apologise."
The demonstrators, organised by Russian youth groups, were teenagers or in their early twenties.
"The attack was probably aimed at me, but as you see nobody came near enough, nobody touched me, everything is OK," said Kaljurand, the ambassador.
Pro-Kremlin youth activists have staged protests outside the Estonian embassy since last week when Estonia's government removed the Red Army memorial from the centre of Tallinn, in an action that has angered many Russians.