* Moscow complains situation worse since Ukraine election
* Demand for U.N. resolution on end to violence
* Fierce gun battle raging in east
* Kiev gets a week's breathing space in gas dispute (Wraps stories on gas dispute, fighting, Lavrov comments)
By Thomas Grove and Mark Trevelyan
DONETSK, Ukraine/MOSCOW, June 2 (Reuters) - Russia accused Ukrainian authorities on Monday of escalating violence against civilians in the rebel-held east of the country, even as it offered Kiev a brief respite in a dispute over billions of dollars' worth of unpaid gas bills.
In the latest fighting, Ukrainian border guards said a pro-Russian militia had attacked one of their posts with automatic weapons and grenade launchers in the early hours, triggering a battle that was still raging many hours later.
Ukraine and its Western allies accuse Moscow of fuelling the pro-Russian uprising that threatens to break up the former Soviet republic of 45 million people. Russia denies orchestrating the unrest, and says Ukraine's attempts to end it by military force are making the situation worse.
Russian Foreign Minister Sergei Lavrov said Moscow would submit a draft resolution to the United Nations Security Council later on Monday, calling for an immediate end to the violence and the creation of humanitarian corridors to help civilians escape the fighting.
In pointed comments aimed at newly elected Ukrainian president Petro Poroshenko, Lavrov said that Western nations had assured Russia the situation in Ukraine would improve after the May 25 election that brought him to power. Instead of that, he said, "everything is happening in exactly the opposite way".
"People are dying every day. Peaceful civilians are suffering more and more - the army, military aviation and heavy weapons continue to be used against them," Lavrov told reporters in Moscow.
Poroshenko and Ukraine's pro-Western government have defied Moscow's repeated calls for an end to what Kiev calls its 'anti-terrorist' operation against armed separatists in the eastern Donetsk and Luhansk regions, who want to follow the example of Crimea by splitting from Ukraine and joining Russia.
The annexation of Crimea prompted the European Union and the United States to impose sanctions on some Russian firms and individuals in the worst standoff between Moscow and the West since the Cold War ended.
On the opposite side of the continent, Poroshenko and Russian President Vladimir Putin will both attend a series of events in France this week to mark the 70th anniversary of the D-Day landings that opened the Western front against Nazi Germany in World War Two.
While no formal meeting between the two is scheduled, even a handshake would be significant. Moscow refused for months to recognise the Ukrainian leadership that replaced its ally, Viktor Yanukovich, when he was toppled by protests in February.
Poroshenko is due to be inaugurated on Saturday and will immediately face an array of crises, including a new deadline in Ukraine's long-running dispute with Russia over gas prices.
On Monday, Russia's Gazprom gave Ukraine an extension into next week to resolve the price question. It had previously said it would switch off the gas on Tuesday unless Kiev agreed to start paying in advance - a step that might also have hit supplies to European countries via Ukraine.
Europe gets a third of its gas needs from Russia, and almost half of these supplies are sent via Ukraine.
Since Yanukovich's overthrow, Russia has demanded a sharp increase in the price Ukraine pays for gas. Kiev says it cannot afford it and wants to pay a discounted price which it negotiated in the past.
While the dispute has dragged on, Gazprom has continued billing Kiev at the higher rate. It says Ukraine already owes it more than $5 billion in unpaid bills and is running up more debt at a rate of more than $1 billion per month.
But after Kiev paid off $786 million of its gas debt, Gazprom announced a six-day extension of the deadline until June 9. Gazprom also said that it would not sue Ukraine's gas supplier Naftogaz over unpaid bills during the coming week.
Talks between the Russian gas exporter and Ukraine were resuming on Monday in Brussels, under the auspices of the European Union.
Despite a pullback of some of the tens of thousands of Russian troops on the border with Ukraine, violence increased in the east of the country at the start of last week, with dozens of pro-Moscow rebel fighters killed in a government assault. Many were Russians, whose bodies were sent back across the border.
In Monday's fighting, security sources said a force of separatists had occupied the upper floors of an apartment block and were shooting into the border post on the southern edge of Luhansk, a city very close to the frontier with Russia.
"Shooting is continuing. There has been no let-up in firing for seven hours now," border post spokesman Oleh Slobodin said.
"We have eight or nine wounded. The attackers have five dead and eight wounded."
In Geneva, a spokesman for the U.N. Office for the Coordination of Humanitarian Affairs (OCHA) reacted cautiously to Russia's proposal to create humanitarian corridors for civilians, saying a crucial question was who would secure them.
"How is it going to be policed? That would be a key question for us," spokesman Jens Laerke said.
"Because once you say here's a corridor, once people start moving on that, if there's no one to protect them, then it's very dangerous." (Additional reporting by Tom Miles in Geneva; Vladimir Soldatkin, Steve Gutterman and Katya Golubkova in Moscow; and Richard Balmforth in Kiev; writing by Mark Trevelyan, Editing by Timothy Heritage and Philippa Fletcher)