Early exit polls appear to give the Georgian billionaire Bidzina Ivanishvili a victory in Georgia's parliamentary election , though his opponent, President Mikheil Saakashvili, also insisted last night that his party was in the lead.

Two polls on Monday put Ivanishvili's Georgian Dream opposition coalition ahead of Mikheil Saakashvili's ruling United National Movement (UNM) party. Imedi TV said Ivanishvili had won 51% of the vote compared with 41% for UNM, with 35% of people refusing to reveal how they had voted. The Georgian public broadcaster had Georgian Dream and UNM tied, with 33% each. Georgian Dream's own pollster reported an overwhelming lead, with 62% against the UNM's 28%.

Speaking after the polls were released, President Saakashvili conceded that the opposition had won in the "proportional race" – a reference to 77 seats chosen on a party-list system. However, he claimed his party was "in the lead with a large margin" outside the capital, Tbilisi. It had a "significant advantage" in 73 constituency seats, he said.

Ivanishvili last night said his party had decisively won the poll. He also said it was on course to win at least 100 seats. He tweeted: "The people of Georgia need to react to our victory calmly and remember that those who voted for the UNM are our brothers."

His prediction – if confirmed on Tuesday – would be a devastating defeat for President Saakashvili, who swept to power in the rose revolution of 2003. It would also mark a remarkable democratic breakthrough for the small post-Soviet state. Whatever the final outcome, Georgia is now likely to have a parliament made up of genuine opposing forces.

In Tbilisi, an opposition stronghold, Ivanishvili supporters poured on to the streets to celebrate. Several thousand celebrated in Freedom Square, waving blue Georgian Dream flags, with cars hooting horns. "The reality is the majority of people in this country support Ivanishvili," said Tarel Peradze, an engineertold the Guardian.

Speaking to his supporters at his party's HQ, Ivanishvili said his main priority was to "unite our country so what was happening in Georgia will never happen again", a reference to Georgia's alleged authoritarian drift during Saakashvili's eight years in power. He restated his commitment to a pro-western foreign policy, saying: "We will move towards Nato and I believe we will become a member of Nato."

The tycoon also said his first foreign trip – assuming he becomes prime minister – would be to the United States. He pledged not to clamp down on critical journalists – a hallmark, according to government critics, of Saakashvili's misrule – and said he would respect Georgia's ethnic minorities.

One senior Georgian government source confirmed Saakashvili had lost the popular vote. But he said it was still possible the UNM would emerge with a "slim majority" in the new parliament. They said Saakashvili's apparent defeat in a fair election showed that the opposition charge he had turned into a despot was evidently erroneous.

Saakashvili is due to carry on as president until October 2013. After this, Georgia shifts from a presidential to a parliamentary system with an executive prime minister in charge. If Ivanishvili does win a parliamentary majority, it is difficult to see how the two men could work together. Ivanishvili yesterday said he was ready to co-operate if made prime minister. "From my side, no problem," he said.

The campaign has featured vituperative statements from both sides, with Ivanishvili hinting in an interview with the Guardian on Sunday that the president was willing to do anything to stop him.