Russia says it is "open to dialogue" with the new president of Ukraine, as initial results suggested Petro Poroshenko would win its election.
However, Russian Foreign Minister Sergei Lavrov said military action must end against separatists in the east.
Mr Poroshenko said he would meet Russian leaders soon but vowed to take a tough line on any armed separatists.
Unrest continues in the east, with pro-Russia militiamen halting flights at Donetsk airport.
Heavy gunfire is now reported there, with thick black smoke rising from the area. Military aircraft have been seen overhead.
Meanwhile election observers said Sunday's vote was a genuine one that largely met international standards.
The mission from the Organisation for Security and Co-operation in Europe (OSCE) also said it gave the new president "legitimacy" to open a dialogue with separatists in the east.
Mr Lavrov told a press conference in Moscow: "We are ready for dialogue with Kiev's representatives, with Petro Poroshenko."
Mr Lavrov said EU and US mediation were not needed, but warned Kiev that continuing military operations against the separatists would be a "colossal mistake".
He said: "As our president [Vladimir Putin] has said, we shall treat the results of the expression of will of the Ukrainian people with respect."
But Mr Lavrov said Kiev must also treat its people with respect and that dialogue with the east was necessary to resolve the crisis.
Mr Poroshenko said he hoped to meet Russian leaders early next month, after a trip to Poland where he will meet the US president and EU leaders.
However, he warned he would take a tough line on armed militiamen.
He said: "Their goal is to turn Donbass [east Ukraine] into Somalia. I will not let anyone do this to our state and I hope that Russia will support my approach."
Mr Poroshenko also indicated he would keep current Prime Minister Arseniy Yatsenyuk, saying: "There are no plans to change the government leadership."
Mr Poroshenko, 48, currently has 53.75% of the vote, with 70% of the ballots counted, and would not need a run-off. Former Prime Minister Yulia Tymoshenko is a distant second on 13.1%.
Full results are expected on Monday.
The election came three months after pro-Moscow President Viktor Yanukovych fled Kiev amid bloody street protests and calls for closer ties with the EU.
Since then, Russia has annexed the Crimean peninsula in southern Ukraine and armed separatists in the eastern provinces of Luhansk and Donetsk have declared independence.
Mr Yanukovych has also said he will accept the election result, "no matter which regions and what percentage of the population voted", Russian media reported.
Ukraine's interim government is engaged in an offensive in the east to quash the uprising that has left scores dead.
Pro-Russian separatists severely disrupted voting there. No polling stations were open in Donetsk city, and across the region only seven out of 12 district electoral commissions were operating.
"Has Ukraine been reborn?" asks the Ukrainian edition of Komsomolskaya Pravda. It says the election "will have a special place in the history books". But the paper notes that Crimeans did not vote and the voting in Donbass and Luhansk - formerly Ukraine's "electoral core", proceeded with "great difficulty".
"We have survived," proclaims popular Ukrainian daily Segodnya.
Claims in the Russian media that Ukraine is overrun by extremists and neo-Nazis are ridiculed by some social media commentators. The two right-wing candidates - Dmytro Yarosh and Oleh Tyahnybok - polled less than 2% in total.
Prominent Ukrainian journalist Mustafa Nayyem proclaims an "epic fail" of Russian propaganda. "Vladimir Putin won't sleep tonight," he predicts.
However, the central elections commission said about 60% of Ukraine's 35.5 million eligible voters had turned out.
Unrest continued on Monday.
Flights were suspended at Sergei Prokofiev airport in Donetsk after several dozen separatists from the self-declared Donetsk People's Republic demanded Ukrainian troops guarding the inner perimeter be removed.
Officials said there had been "shots and confrontation".
The head of Donetsk airport's press service, Dmitry Kosinov, said: "On the territory of the airport there are armed people, this is ample reason to halt our work on security grounds."
Ukrainian TV is also reporting clashes in the villages of Semenivka and Andriyivka near the separatist stronghold of Sloviansk.