By Margarita Antidze and Steve Gutterman

TBILISI (Reuters) - Georgia's President Mikheil Saakashvili on Tuesday conceded defeat in parliamentary elections to a coalition led by a tycoon promising to ease tensions with Moscow, four years after the staunch U.S. ally lost a war with Russia.

Russian Prime Minister Dmitry Medvedev, who has long been openly hostile to Saakashvili, welcomed the opposition victory as opening the way for "more constructive and responsible forces" to enter the Georgian parliament.

Saakashvili's acceptance that his ruling party will go into opposition to Bidzina Ivanishvili's Georgian Dream increased the chances of the country's first peaceful transfer of power between rival parties since the Soviet Union collapsed in 1991.

Although it strengthens Georgia's democratic credentials, it could lead to an uneasy cohabitation between Ivanishvili, who is likely to become prime minister, and Saakashvili, who does not step down as president until next year.

Instability in the country would worry the West because it is a conduit for Caspian Sea energy supplies to Europe and has a strategic location on the Black Sea between former Soviet master Russia and Iran, Turkey and central Asia.

Tonino Picula, who led a team of international observers monitoring the vote, endorsed the conduct of the elections.

"Despite a very polarizing campaign that included harsh rhetoric and shortcomings, the Georgian people have freely expressed their will at the ballot box," he said in a statement.

Medvedev, who was president and commander-in-chief when Georgia fought its disastrous war with Russia in 2008, expressed hopes that the result would improve the relationship between the countries.

"We can only welcome this as it probably means that more constructive and responsible forces will appear in parliament," Russian news agencies quoted him as saying.

Russia has had no diplomatic relations with Georgia since the five-day war, and both Medvedev and President Vladimir Putin have refused to have any personal contacts with Saakashvili, whom they depict as a dangerously aggressive hothead.

The agencies said Medvedev was speaking in his capacity as leader of the United Russia party, and Putin has yet to comment.

The U.S. State Department congratulated Saakashvili for "graciously conceding" and said it sent a good signal.

Georgian Dream supporters celebrated their victory in the capital on Monday in scenes reminiscent of the euphoria of the 2003 Rose Revolution that propelled Saakashvili to power.

Motorists were again driving through Tbilisi on Tuesday evening with flags flying and horns sounding.

TYCOON SET TO BECOME PREMIER

Defeat for Saakashvili's United National Movement (UNM) followed criticism that he has monopolized power, mistreated opponents and trampled on rights and freedoms. Video footage of torture, beatings and sexual assault of prison inmates, aired by opposition channels before the election, had led to protests.

"Justice has been restored," said Nino Kantaria, 42, in Tbilisi. "I believe that Bidzina will make our lives better."

But Zaira Khabuliani, another resident of the capital, said she was unsure of the opposition's ability to govern.

"I don't know how Ivanishvili will behave and what he will do for people."

Ivanishvili, a once reclusive businessman who built his $6.4 billion fortune in Russia, said he was confident of becoming prime minister.

Saakashvili had said Georgian Dream would move the country of 4.5 million away from the West and back into Moscow's orbit, suggesting Ivanishvili would do the bidding of the Kremlin.

"We'll do our best to sort out relations with Russia," Ivanishvili, 56, told reporters, but added: "Our main aspiration is Europe and our security is NATO."

In Tbilisi, two U.S. senators met Ivanishvili and congratulated the Georgian people on the election.

"We discussed at length how important the relationship is between Georgia and the United States of America, and we look forward to ... continuing to build that relationship," Republican Senator Jim Risch of Idaho said.

"We are good friends, we want to continue to be good friends," he said, standing beside Ivanishvili.

Governing Georgia could be much harder as, until Saakashvili's term ends next year, he will no longer have a compliant parliament and he will have to work with Ivanishvili.

Saakashvili accepted the will of the majority but said some of Georgian Dream's views were "fundamentally unacceptable" and saw "very deep differences" with the coalition.

Under reforms that take effect after a presidential election next year, the authority of the head of state will be weakened and more power will go to parliament and the prime minister, who will become the most powerful executive official.

Ivanishvili set out plans he would pursue as premier, saying a balanced budget would be a priority.

U.S.-educated Saakashvili curbed corruption and presided over an economic resurgence, but the war set back efforts to bring Georgia into NATO and gain control over the Moscow-backed breakaway regions of South Ossetia and Abkhazia.

The economy has grown again after contracting in 2009, hit by the war and the global economic crisis, but the official unemployment rate of 16 percent is considered to be an underestimate, and many Georgians struggle to pay bills.

"Of course, there have been problems. High utility tariffs, unemployment," said Alexander Begiashvili, who backed Georgian Dream. "A man lives only once and wants to have a good life."

(Additional reporting by Nino Ivanishvili; Writing by Steve Gutterman, Editing by David Stamp and Will Waterman)