Russian Foreign Minister, Sergei Lavrov, said that military action must end against separatists in the east.
Mr Poroshenko said that 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.
Meanwhile election observers said Sunday’s vote was a genuine one that largely met international standards.
The mission from the Organization 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.
Petro Poroshenko is a confectionery tycoon who backed the pro-European street protests that ousted Moscow ally Viktor Yanukovich from Presidency.
He was the only Ukrainian strongman to swiftly throw his weight behind the popular uprising that started late 2013, devoting his ‘5th Channel’ TV station to the coverage of the protests.
A seasoned politician who served as Foreign Minister and Economy Minister in previous administrations, Poroshenko’s chain of confectionery shops has put the billionaire on Ukraine’s top 10 rich list, earning him the nick-name “Chocolate King”.
Poroshenko, whose chocolate manufacturer, Roshen is one of the world’s top 20 confectionery firms, backs Ukraine’s integration with the west and his firm has been targeted in Russia.