David Cameron told Vladimir Putin he must ensure access to the MH17 crash site so victims of the disaster can hold proper funerals for their loved ones.
The Prime Minister spoke to the Russian president for the first time since MH17 was brought down, apparently by pro-Moscow rebels, in eastern Ukraine on Thursday.
Mr Cameron, who was left frustrated by Mr Putin's refusal to speak to him until today, said he had "made clear" his demands in the phone call which came as the European Union looked set to impose tougher sanctions on Russia over its support for rebels in Ukraine.
The Prime Minister said: "I've just spoken to president Putin. I made clear he must ensure access to the crash site so the victims can have proper funerals."
Rebels in eastern Ukraine took control of the bodies recovered from the downed Malaysia Airlines aeroplane.
Donetsk rebel leader Alexander Borodai said the bodies recovered from the crash site would remain in four refrigerated train cars in the rebel-held town of Torez, nine miles from the crash site, until the arrival of an international aviation delegation.
A No 10 spokeswoman said Mr Cameron had made clear to Mr Putin that the shooting down of MH17 was "totally unacceptable" and Russia's failure to cease support for the separatists had "contributed to an appalling tragedy".
Mr Cameron told the president that the "world was now watching" and he "must change course and work to bring stability to eastern Ukraine", the spokeswoman said.
She said: "The Prime Minister spoke to president Putin this evening and made clear that the shooting down of MH17 was totally unacceptable.
"The evidence suggested that pro-Russian separatists were responsible and the Prime Minister made clear that if Russia wants to put the blame elsewhere they would need to present compelling and credible evidence.
"The PM made clear that our priority is to get experts to the crash site so they can recover and repatriate the victims and collect any evidence necessary for the investigation.
"The PM emphasised that the families of 298 individuals need to know that everything is being done to make this happen and called on president Putin to use his influence on the pro-Russian separatists to ensure this happens. The delay and restrictions so far were completely unacceptable and indefensible.
"The PM said that the tragedy had brought into sharp focus the consequences of destabilisation in eastern Ukraine.
"The G7 and EU had repeatedly called on president Putin to cease support for the separatists and to work with the rest of the world to find a peaceful resolution. Russia's failure to do so had contributed to an appalling tragedy."
Mr Cameron told the Russian leader that EU foreign ministers would meet on Tuesday to "consider further sanctions on Russia given its inaction".
Foreign Secretary Philip Hammond accused Russia of "obfuscation and obstruction" and said Mr Putin could "snap his fingers" and allow a proper investigation to take place at the crash site, but that had not been done.
He warned Russia could become a "pariah state" if it did not behave properly on the international stage.
Mr Hammond told BBC1's Andrew Marr Show: "The eyes of the world are on Vladimir Putin and what we are seeing from the Russians is obfuscation and obstruction at the moment.
"The Russians will have probably more information about this incident than anyone. They are very close by, this is only a few miles from the Russian border, they have got lots of military planes in the area, they are saying nothing.
"What we need is full Russian co-operation. Any evidence they control needs to be turned over to the international investigators. They must use their influence to allow international access to the site, and to secure the evidence and to secure respect for the bodies and the possessions of the victims."
He added: "There is one party in the world who clearly has the ability to snap his fingers and it would be done, and that's Vladimir Putin, and for all the fine words we are hearing from Moscow it hasn't happened."
Mr Hammond, who has chaired a series of meetings with Whitehall officials including representatives of the intelligence agencies, said the evidence available about those who were behind the atrocity was not yet strong enough to stand up in court, but it would "lead the reasonable person to the unavoidable conclusion that this was a missile fired from rebel-held territory, almost certainly a missile supplied by the Russians".
On Sky News' Murnaghan programme, Mr Hammond added: "This is about Russia and the entire international community, and Russia risks becoming a pariah state if it does not behave properly."
Two officers from the Metropolitan Police have arrived in Ukraine to assist in the grim task of recovering, identifying and repatriating the bodies of those killed.
A team of six investigators from the Air Accidents Investigation Branch is already in the country, working with international counterparts on the next steps in the effort to establish what happened to MH17, although Downing Street said there were still concerns about getting access to the crash site.
All 298 people, including 10 Britons, on board the flight from Amsterdam to Kuala Lumpur were killed.
Earlier, tougher European Union sanctions on Russia moved a step closer after talks between David Cameron and the leaders of France and Germany.
Downing Street said the Prime Minister, German Chancellor Angela Merkel and French President Francois Hollande agreed that EU foreign ministers should be ready to impose further sanctions on Russia when they meet on Tuesday.
The agreement from three of the leading EU members that foreign ministers should be ready to impose further sanctions follows frustrations in Westminster about resistance from Germany to a more punitive regime.
Mr Hammond indicated the sanctions could go right to the heart of Mr Putin's inner circle, and could include restrictions on the supply of arms to Russia - a measure which the UK has already adopted but has so far been resisted at an EU level.
One option was "broadening the number of individuals who are subject to sanctions to include the so-called crony group around president Putin", he told BBC Radio 4's The World This Weekend.
US secretary of state John Kerry said evidence was "stacking up" about the incident which Russia "needs to help account for".
He said: "We are not drawing the final conclusion here. But there is a lot that points at the need for Russia to be responsible."
Mr Cameron also spoke to Australian counterpart Tony Abbott about a new United Nations security council (UNSC) resolution which would guarantee "unfettered access" to the crash site for experts.
Russia blocked a UNSC press statement on the issue over the weekend, and No 10 would not be surprised if it vetoed the resolution tomorrow, but it forms part of the international strategy to increase pressure on Mr Putin.