German Foreign Minister Frank-Walter Steinmeier, hosting the six original states of the European Union in Berlin, said they were in agreement that London must not wait to start the complex procedure of extracting itself from the bloc.
"We join together in saying that this process must begin as soon as possible so we don't end up in an extended limbo period but rather can focus on the future of Europe and the work toward it."
His French counterpart Jean-Marc Ayrault said it was urgent David Cameron, who on Friday said he would resign by October, clear the way for new leadership to manage the transition out of the union.
"A new prime minister must be designated, that will take a few days but there is a certain urgency," he told reporters at the crisis talks after Britain's shock referendum to quit the EU.
In announcing his resignation by October, Cameron said it should be his successor who leads the negotiations under Article 50 of the EU's Lisbon Treaty which sets out a two-year timeframe to leave.
However Steinmeier, Ayrault, the Netherlands' Bert Koenders, Italy's Paolo Gentiloni, Belgium's Didier Reynders and Luxemburg's Jean Asselborn stressed in the German capital that time was of the essence.
"We understand and respect the result (of the referendum) and understand that Britain is now concentrating on Britain," Steinmeier said, flanked by the other ministers.
"But London has a responsibility toward more than just Britain. We must now be allowed to focus on the future of Europe and that means that after the decision taken in Britain, the process by which we negotiate Britain's exit must begin."
Koenders called for "good faith" talks with London to begin immediately: "We have to move on... we need to turn the page."
- 'Enormous challenges' -
Steinmeier had opened the meeting with a heartfelt plea for the EU to remain united.
"I am confident that these countries can also send a message that we won't let anyone take Europe from us," he said.
Later in a joint statement, the ministers defended the EU's "long and successful" path since its humble beginnings in 1957.
But they acknowledged that the union needed to address the fact that "parts of our societies are not happy with how the EU works at the present time".
"The European Union stands before enormous challenges in a globalised world, which only a better European Union can grapple with," they wrote, citing migration and refugees, security, economic growth and jobs as key priorities.
They also acknowledged that among the remaining 27 member states there were differing "levels of ambition" for European integration and had "to find ways to better deal with" them.