Prime minister David Cameron has called for Greece to move "decisively and swiftly" to form a new government, warning that "delay could be deadly".
Mr Cameron was speaking on Monday at a G20 summit in Mexico which was dominated by the urgent need for a resolution to the crisis in the eurozone.
The prime minister said that, by backing the centre-right New Democracy party in Sunday's elections, the Greek people had "made their choice to stay in the euro" and should now form a government prepared to deliver on promises made in return for multibillion-euro bailouts.
But he also called on the other eurozone countries - and particularly Germany and the European Central Bank - to take "bold steps" towards the fiscal and banking union which he believes are necessary to solve the underlying problems with the single currency, the Press Association reported.
"The Greek people have now made their choice to stay in the euro, to accept the consequences of what that involves," said Mr Cameron. "Those parties that believe that need to get into government, need to deliver that.
"Delay is always dangerous in these situations. It is in our interests in Britain that these issues are resolved decisively and swiftly and that is what we are urging people to do today."
German chancellor Angela Merkel's spokesman said she regards the Greek election results as "good news for the euro, for the eurozone and for Europe".
But Ms Merkel indicated that she is not willing to countenance any relaxation in the demands placed on Athens for austerity measures in return for its bailout.
"The important thing is that the new government sticks with the commitments that have been made," she told reporters at the summit in the Pacific resort of Los Cabos. "There can be no loosening on the reform steps."
Germany is coming under intense pressure at Los Cabos to use its financial clout to underwrite a plan to stabilise the single currency.
In a speech to a business audience ahead of the official opening of the G20, Mr Cameron said: "The reality is that there are a set of things that eurozone countries need to do. And it's up to eurozone countries whether they are prepared to make the sacrifices these entail.
"The challenge is one of political will as much as economics. Of course these things are difficult to do, but just because these things are difficult does not mean we should not say them.
"If the eurozone is to stay together then it has to make at least some of these difficult decisions.
"The alternatives to action that creates a more coherent eurozone are either a perpetual stagnation from a eurozone crisis that is never resolved or a break-up caused by a failure to address underlying economic fundamentals that would have financial consequences that would badly damage the world economy, including Britain."
Asked in a round of TV interviews whether he feared the eurozone crisis could drag on for years, Mr Cameron said: "I very much hope that won't be the case.
"If eurozone countries act decisively, if they act clearly, if they deal not only with the short term financial pressures but also the longer-term underlying problems of the euro, if they take these steps, we can see this crisis ease.
"But the alternatives are not good."
Mr Cameron cautioned against placing all the blame for the failure to make progress in the eurozone on Berlin.
"I can quite understand the concern that the Germans have," he said. "Their taxpayers have been asked to put a lot of money into Greece, they have done that and so they want the Greeks to stick to their side of the bargain. That's what the Germans are saying.
"The truth is that everyone in the eurozone is going to have to take difficult decisions in order to make the system work properly and deliver an easing of the crisis, which we all want to see - and it affects us in Britain too."
Mr Cameron is expected to join Ms Merkel and eurozone leaders Mario Monti, of Italy, and Francois Hollande, of France, with US President Barack Obama later to discuss the way forward for the single currency.
Mr Cameron said that he would be "standing up for British jobs and British business" in Mexico.
The gathering of leaders of the world's major economies was not just a talking shop, but the right forum to deal with the five big threats to the world economy - the eurozone crisis; sovereign debt; protectionism; low growth; and the failure to regulate banks.
"It is important not just to talk, but to deal with the risks that face the world economy," said Mr Cameron.
"We could make a difficult situation worse if we fail to tackle the eurozone crisis, and then add to that a slide towards protectionism and not meeting the commitments we have made on banking regulation.
"I am determined that is not going to happen. The G20 has a purpose that directly relates to people back at home, because we will all be better off if we sort out the eurozone crisis, if we stabilise and make safe our banks and financial services and make sure the world economy is open for trade.
"Britain is a huge trading nation. Even here in Mexico we are the fifth biggest investor into Mexico.
"We rely on the world trading system being open and organisations like the G20 can help that, and that means jobs and growth and wealth and investment back home."