German Foreign Minister Frank-Walter Steinmeier said negotiations should begin as "soon as possible".
He made the comments after an urgent meeting of the six EU founder members to discuss the decision.
British PM David Cameron has said he will step down by October to allow his successor to conduct talks.
The six countries attending the summit in Berlin - Germany, France, Italy, Belgium, Luxembourg and the Netherlands - first joined forces in the 1950s and still form the core of the EU.
"We say here together, this process should get under way as soon as possible so that we are not left in limbo but rather can concentrate on the future of Europe," Mr Steinmeier said.
His Dutch counterpart Bert Koenders said the continent could not accept a political vacuum, saying "this will not be business as usual".
Meanwhile on negotiations, German Chancellor Angela Merkel said: "It shouldn't take forever, that's right, but I would not fight for a short time frame."
She added that she was seeking a "objective, good" climate in talks on Britain's exit from the EU, and that there was no need to make deterrence a priority.
Mrs Merkel also said there was "no need to be particularly nasty in any way in the negotiations; they must be conducted properly".
In other developments:
The first summit of EU leaders with no British representation will be held on Wednesday, a day after Mr Cameron holds talks with members.
Global stock markets and the pound fell heavily on the news of the so-called "Brexit", while credit rating agency Moody's cut the UK's outlook to "negative".
The UK must now invoke Article 50 of the EU Lisbon Treaty, which sets out a two-year timetable for negotiations on withdrawal.
European Commission head Jean-Claude Juncker said the EU-UK split was "not an amicable divorce", but nor had they had a "deep love affair". He has also said exit negotiations should begin immediately.
"Britons decided yesterday [Thursday] that they want to leave the European Union, so it doesn't make any sense to wait until October to try to negotiate the terms of their departure," Mr Juncker said in an interview with Germany's ARD television network.
"OUTsch!," "Well done little Britain," "Completely detached."
Germany's press reflects the widespread shock at the British decision to leave the EU. At the highest level of government no-one really expected this. And they are scrambling to defend German and European interests.
The German position is focused on two major, and delicate, decisions.
Firstly, whether to use Brexit as an opportunity to deepen European integration, or to view it as a "wake-up call" and create more flexibility within the union.
And, secondly, how to deal with Britain - an important trading partner - as a 'third country.'
Senior economists and business leaders warn against barriers to free trade.
But politicians are talking tough. Concessions, they say, might encourage other member states to leave. For this reason one senior MP told me "there must be consequences for Britain".
Other leaders, including Polish President Andrzej Duda, said it was crucial that work be done to prevent other countries leaving the EU.
Turkey's President Recep Tayyip Erdogan said it was inevitable more countries would leave the EU soon if it did not change its path.
Leaders of Eurosceptic parties in France, the Netherlands and Italy quickly demanded referendums in their own countries.
Brexit: What happens now?
EU states have urged Britain to hold speedy talks on leaving the bloc after it voted to end its membership in a historic referendum.