The prize was awarded by Parliament's President Jerzy Buzek who drew on his own experience in Communist Poland to talk about the significant of human rights. He also said he was "particularly proud" to chair the Parliament and thus be able to award this prize which had been voted for a majority of Members.
Mr Buzek also spoke of murdered activists Natalia Estemirova and Anna Politkovskaya whom he said "should be here today: their killers have to be brought to justice".
By an ironic coincidence not lost on those in the Chamber, Russian Memorial was founded by Andrei Sakharov to document Soviet crimes.
"Murdered in Moscow, shot in St Petersburg"
Accepting the prize, Sergei Kovalev spoke on behalf of the group. "I am sure that, in awarding the Sakharov Prize to Memorial, the European Parliament had them in mind, first and foremost - our dead friends, comrades-in-arms, kindred spirits. This prize belongs by right to them. And the first name I should cite is that of Natalya Estemirova, human rights defender and fellow member of Memorial, murdered this summer in Chechnya."
Mr Kovalev said, "I cannot go on without mentioning other names too: the lawyer Stanislav Markelov and journalists Anna Politkovskaya and Anastasia Baburova, murdered in Moscow, ethnologist Nikolai Girenko shot in St Petersburg, Farid Babayev, murdered in Dagestan, and many others - sadly, it is a list that could go on for a long time."
Mr Kovalev then invited Members to stand for a moment to honour those who had died.
Tribute to Andrei Sakharov "a distinguished thinker"
Mr Kovalev said "Andrei Sakharov, who died 20 years ago, was not just a distinguished champion of human rights in the Soviet Union. He was also a distinguished thinker".
"The European Community, whose Parliament instituted this prize is, perhaps, today the model closest to that future united humanity dreamt of by Andrei Dmitrievich Sakharov".
On Russia: "Today my country needs support and pressure"
"The situation in our country is not as straightforward as it might appear to the superficial observer. We have many allies in society - both in our struggle for human rights and in our struggle with Stalinism. Moreover, Russian authority is also not as homogeneous as it might seem at first sight," he said.
"What can we expect here from European politicians and from European public opinion?" he asked. "Andrei Sakharov formulated these expectations more than 20 years ago: "'Today my country needs support and pressure'"
He elaborated, saying "so what should Europe do in relation to Russia? It should act towards Russia just as it does towards any other European country that has taken on certain obligations and has a responsibility to meet them. It is Europe's duty not to remain silent but, again and again, to repeat and remind, and insist respectfully and firmly that Russia meets its obligations," he said.
"The basis of all other freedoms"
Mr Kovalev finished by saying, "freedom of thought is the basis of all other freedoms. That is why it is so appropriate for the Sakharov Prize to be called 'For Freedom of Thought'. We are proud to receive it today."