Story highlights Rebel Prime Minister wins in self-proclaimed People's Republic of Donetsk

Russian Foreign Ministry considers the vote valid

Exit polls show more than 500,000 people voted in Luhansk

The Ukrainian government and Western countries have criticized the elections

Voters in rebel-held eastern Ukraine turned out at the polls Sunday to elect members of Parliament and prime ministers in the pro-Russian separatist provinces of Donetsk and Luhansk in a vote criticized by the Ukrainian government in Kiev and Western governments.

Against a backdrop of violence and bloodshed, the vote played out despite a truce on paper between pro-Russian separatist groups and Ukrainian government forces.

Moscow has said it will recognize the results, while the central Ukraine government in Kiev, the United States and European Union have condemned the elections, calling them "illegitimate" and a violation of the Minsk agreement.

The self-proclaimed People's Republic of Donetsk announced that its current Prime Minister, Aleksandr Zakharchenko, had won the vote there.

After all the ballot papers were counted, Zakharchenko secured 765,350 votes, Roman Lyagin, the leader of the Central Election Commission in Donetsk, told a news conference Monday. He didn't say what percentage of the vote that was.

"Everything went better than expected," Lyagin had earlier told Novorossiya, the press center for the separatist Republics of Donetsk and Luhansk.

In the self-declared Luhansk People's Republic, the head of the Central Election Commission said more then a half million people turned out to cast ballots at over 100 polling stations, including three inside Russia, according to preliminary numbers from the CEC's website. The vote count is now under way.

Earlier Sunday, Luhansk CEC head Sergei Kozyakov told Itar-Tass so many people turned out to vote they extended polling station hours at several locations until 10 p.m.

The Russian news agency also reported on a statement from the Russian Foreign Ministry, which said it considers the vote valid.

"The elections in the Donetsk and Lugansk regions were held in an organized way in general and with high voter turnout," the ministry said, according to Itar-Tass. "We respect the will expression of the citizens of the southeast."

Authorities in Kiev are basically ignoring the election. In a statement Sunday, the Security Service of Ukraine said "it is a false election ... and a farce organized by criminals to collect data about people."

Before the vote, Ukrainian President Petro Poroshenko called the ballot "pseudo elections ... by terrorists and bandits."

Ukrainian authorities also said Sunday they are launching a criminal investigation into the separatist elections, calling it "a power grab," Agence France-Presse reported, citing the Security Service of Ukraine.

Vegetables in exchange for votes

Some residents of the devastated region have said they were determined to participate in the vote, whether it's recognized by Kiev or not.

And many were rewarded with offers of free or cheap fruits and vegetables. Pictures at polling stations show large piles of vegetables outside stations in some areas in a bid to lure voters to polls.

Produce aside, people said they hoped the election would help end the violence.

"Day and night we hear shots. Yes, they bomb us. Yes, we have bomb shelters and basements. But we don't want to go back there. That's why we are preparing for the elections and hope for something better," said Tatyana, a schoolteacher.

"Mostly people are afraid. The constant shooting scares people. But of course we should go and vote to reach something good in life," said Valentina.

But there is no way to even verify the credibility of the balloting in the rebel-held areas. The voting has actually been under way for a week, by email and at mobile voting stations. And no official observers have been used to monitor the voting.

Last weekend, the bulk of Ukraine under government control held its own parliamentary elections , which kept pro-Western parties in power.

But citizens in Russian-annexed Crimea and the eastern areas controlled by separatists didn't participate in those elections because of the ongoing violence.

U.S. and Russia at loggerheads

As Ukraine wrestles with internal turmoil, torn between Russiaan influence and Western support, the United States slammed the separatist vote.

"US will not recognize any election in separatist areas that doesn't follow Ukrainian law (and) is not held with consent of Ukrainian government," Samantha Power, U.S. ambassador to the United Nations, tweeted on Sunday.

"We deplore the intent of separatists in parts of eastern Ukraine to hold illegitimate so-called local 'elections' on Sunday, November 2," National Security Council spokeswoman Bernadette Meehan said in a statement Friday.

She warned Russia "against using any such illegitimate vote as a pretext to insert additional troops and military equipment into Ukraine."

Russia is rejecting calls to distance itself from the rebel vote.

"We will of course recognize their results. And we expect that this election will be free and that no one will try to disrupt it from the outside," Russian Foreign Minister Sergei Lavrov said last week.