Viktor Yanukovich and Yulia Tymoshenko, the Ukranian prime minister, took the largest share of votes on Sunday, with both expected to try to improve relations with Russia if they are elected next month.
"I hope the tallying of the final results in Ukraine will lead to a capable, effective government, oriented on the development of constructive, friendly and multi-dimensional relations with Russia," Medvedev said.
Yushchenko, who came to power in the pro-Western Orange Revolution of 2004, had repeatedly clashed with the Russian government.
He angered Moscow by seeking to bring Ukraine into the Nato military alliance, supporting Russia's foe Georgia and campaigning for a Stalin-era famine that killed millions of Ukrainians to be classified as genocide.
But he won only five per cent of the vote in Sunday's first-round election, according to near-final results, as voters punished him for failing to bring about promised reforms and extract Ukraine from a deep economic crisis.
Sergei Lavrov, Russia's foreign minister, also indicated Moscow's pleasure with Yushchenko's departure from power.
"Regarding the prospects of Russian-Ukrainian relations, we have always been against approaches that are politicised, artificial and have nothing in common with the interests of the Ukrainian people," Lavrov said on Tuesday.
"We hope the new president, whose name we will most likely find out on February 7, will fully understand the need to build relations in this manner and not hold them hostage to his or her ambitions."