* Lavrov warns Kiev over "anti-terrorist" operation
* Sees chance to ease tension in Ukraine
* Criticises vote but does not call it illegitimate (Adds quotes on armed operation, chance for dialogue)
By Steve Gutterman
MOSCOW, May 26 (Reuters) - Russia said on Monday it was ready for dialogue with Petro Poroshenko, Ukraine's next president, but warned the Kiev authorities not to step up armed operations against separatists in the east.
Echoing remarks by President Vladimir Putin in the past few days, Foreign Minister Sergei Lavrov said the key to resolving Ukraine's crisis was genuine dialogue between Kiev and the east - where pro-Russian separatists have rebelled against the national leadership.
"As the president (Putin) has said more than once, we are ready for dialogue with representatives of Kiev, we are ready for dialogue with Petro Poroshenko," he told a news conference when asked about Sunday's presidential election in Ukraine.
Lavrov's remarks signalled that the Kremlin believes acceptance of the election, at least for now, represents Russia's best chance of a role in influencing Ukraine's future.
Poroshenko, a billionaire chocolate manufacturer, won more than 50 percent of votes cast in Sunday's election, preliminary results showed. He had to cross the 50 percent barrier to avoid a two-candidate second-round runoff.
Russian officials for weeks questioned whether the election would be legitimate while the army was being deployed against civilians, and massed troops on the frontier with Ukraine, prompting concern in the West that Moscow would invade a country it regards as the cradle of Russian civilisation.
The crisis, in which Kiev's Moscow-leaning president was ousted and Russia annexed the Crimea peninsula from Ukraine, has become the worst East-West standoff since the Cold War.
But Putin softened Russia's stance before the vote, indicating it would work with whomever becomes president of Ukraine, and analysts in Moscow said he sees Poroshenko as the candidate with whom he was most likely to be able to business.
NO NEED OF MIDDLEMEN
"Taking into account the expression of will that has taken place, which we respect, we will be prepared to establish pragmatic, equitable dialogue on the existing foundation - by which I mean the fulfilment of all existing agreements, including in trade and the gas sector," Lavrov said.
Lavrov did not directly question the legitimacy of the election, in which millions of potential voters in the east were unable to cast ballots, but left the door open for such criticism by saying the campaign was "not without problems".
"Far from all candidates were able to go the distance ... Many had to quit the race, in some cases physically fearing for their lives in the face of threats," Lavrov said.
Lavrov also suggested the vote was marred by the Kiev government's security operation against the separatists, who control buildings in several towns and cities.
"When we speak of how voting went and the election results, we cannot close our eyes to the fact that the so-called counter-terrorist operation was not halted," Lavrov said, adding that stepping up the operation would be a "colossal mistake".
"The chance that now exists to establish mutually respectful, equitable dialogue ... must not be missed."
He said he would "very much not like" Poroshenko to place conditions on dialogue between Moscow and Kiev, such as mediation by the United States and European Union.
"We probably don't need middlemen," Lavrov said, but he added that Russia was ready to work with the EU and United States to help the Ukrainian government and its opponents take steps to ease tension and resolve the crisis. (Additional reporting by Ludmila Danilova, Editing by Timothy Heritage and Peter Graff)