The story, the latest in a series of allegations about US intelligence activity, is likely to further strain relations with its European partners.
European Parliament President Martin Schulz said in a statement he was shocked at the allegations and demanded full clarification from Washington.
Der Spiegel said its report was based on confidential documents, some of which it had been able to consult via Snowden.
It was former intelligence analyst Snowden who earlier this month revealed details of the so-called Prism programme operated by the US National Security Agency (NSA).
He is currently in the transit area at Moscow airport, seeking a country that will accept his request for asylum.
One document, dated September 2010 and classed as "strictly confidential", describes how the NSA kept tabs on the European Union's diplomatic mission in Washington, Der Spiegel said.
Microphones were installed in the building and the computer network infiltrated, giving the agency access to emails and internal documents.
The EU delegation at the United Nations was subject to similar surveillance, Der Spiegel said: the leaked documents referred to the Europeans as "targets".
And the spying also extended to the 27-member bloc's Brussels headquarters.
Der Spiegel referred to an incident "more than five years ago" when EU security experts discovered telephone and online bugging devices at the Justus Lipsius building.
In 2003, the EU announced it had found phone taps in the building targeting the offices of several countries, including Britain, France and Germany. It was not immediately clear if Der Spiegel was referring to this case.
'Out of control'
Schulz, in his statement, said: "I am deeply worried and shocked about the allegations of US authorities spying on EU offices.
"If the allegations prove to be true, it would be an extremely serious matter which will have a severe impact on EU-US relations."
He demanded full and speedy clarification from the US authorities.
US spying was "out of control", said Luxembourg Foreign Minister Jean Asselborn.
"The US would do better to monitor its intelligence services instead of its allies," he added.
Even before the latest allegations, the EU's Justice Commissioner Viviane Reding wrote to US attorney general Eric Holder earlier this month calling for answers about its internet spy programme.
"Fundamentally, this is a question of trust," Reding said in a June 14 speech. "Trust of citizens towards their governments and to the governments of partner nations."
The top-secret Prism programme collects and analyses information from internet and phone users around the world, with access to data from Google, Yahoo! and other internet firms.
US officials say the information gathered is vital in the fight against global terrorism.
The diplomatic squabbling over the fate of Snowden continued, meanwhile.
President Rafael Correa said that US Vice-President Joe Biden had, in a telephone conversation on Saturday, asked Ecuador to reject any asylum request from Snowden.
But in a television interview later, Correa said Snowden's fate was in Russia's hands as Quito could not process Snowden's asylum until he was on Ecuadoran soil.
The president said his government would listen to opinions from US authorities, but that the final decision rested with Ecuador.
Snowden, a former NSA contractor, has had his US passport revoked.
"We have not sought out this situation," said Correa.
"Snowden is in contact with [WikiLeaks founder Julian] Assange, who recommended he seek asylum in Ecuador," he added.
Ecuador has already granted refuge to Australian activist Assange, founder of the anti-secrecy WikiLeaks website.
Assange, who is wanted for questioning in Sweden on sex assault allegations, took refuge at the Ecuadoran embassy in London a year ago to avoid Britain putting him on a plane to Stockholm. – AFP