"Russia's aim was to repeat in Odessa what is happening in the east of the country," Xinhua quoted Yatsenyuk as saying during a visit to Odessa.
"What happened in Odessa was part of a plan by the Russian federation to destroy Ukraine and its statehood," he said. "Russia sent people here to create chaos."
He added that he had told prosecutors to "find all instigators, all organisers and all those that, under Russian leadership, began a deadly attack on Ukraine and Odessa".
Clashes erupted between pro- and anti-govt protesters in Odessa, killing over 40
Ukrainians should unite and put an end to their differences "so as not to give the terrorists backed by Moscow the opportunity to divide our people", the prime minister said.
Yatsenyuk also said that all the city's police chiefs had been sacked for not preventing the clashes and ensuing deaths.
Clashes erupted between pro- and anti-government protesters in Odessa Friday, killing at least 43 people and injuring more than 170 others.
Tensions in eastern Ukraine have been escalating since mid-April following an armed confrontation between Ukrainian security forces and pro-Moscow activists, who demand a referendum on autonomy and closer ties with Russia.
Kiev has repeatedly blamed Moscow for inciting the unrest and splitting Ukraine. Russia denies the charges.
In Moscow, Russian Deputy Foreign Minister Grigory Karasin said Sunday that Kiev needed help to establish a dialogue with pro-Russia activists in southeastern Ukraine.
"It appears that without external help the Kiev authorities are not capable of establishing such a dialogue," Karasin told Rossiya-24 television.
He said steps would soon be taken to bring a dialogue about, without giving further details.