A projection, based on 29 percent of votes counted, gave the Socialist Party of former Tirana mayor Edi Rama 84 of parliament's 140 seats, ahead of Berisha's Democrats on 56.

If confirmed, the result would end eight years of rule by Berisha and bring to power 48-year-old Rama, an artist and Communist-era rebel who won international acclaim for revitalising Albania's drab capital with splashes of paint and avenues of trees.

The European Union and Albania's allies in NATO want an orderly transition, anxious to avoid confrontation in a country that is no stranger to political unrest.

Rama urged Berisha, Albania's dominant political figure since the collapse of more than four decades of Stalinist rule in 1991, to step aside.

"There always comes a time to lose, and today is the time to do it for the honour of Albania," he said at his party headquarters.

"Albania must emerge from these elections holding its head up high," he said. "This is the moment in politics when the losers can take part in the victory of their country."

Like the Socialists, Berisha's party also declared victory within minutes of polls closing on Sunday, raising the spectre of a disputed result that would further bog down Albania's stalled bid to join the EU.

"FULL CONFIDENCE"

There was no word from Berisha himself on Monday but his Democrats stuck to their guns.

"We continue to have full confidence we shall be the winners when all the votes of the citizens have been counted," said Democrat lawmaker Gerti Bogdani.

Berisha is bidding for an unprecedented third successive four-year term. At 68, defeat could mean the end of his career.

A fiery former cardiologist, Berisha took Albania into NATO and onto the first rung of EU membership but his opponents accuse him of undermining democracy and allowing graft and organised crime to flourish.

Rama, a towering former basketball player, says he will reboot Albania's bid to join the European Union and transplant his success in Tirana to the rest of the rundown country.

He has talked of introducing a progressive tax rate and easing the burden on small businesses.

Since 1991, the impoverished country of 2.8 million people has never held an election deemed fully free and fair.

The EU said this vote was a "crucial test" of its progress towards the 27-nation bloc, which Croatia will join on July 1.

Albania applied to come aboard four years ago but has not yet been made a candidate for membership due to concerns over the state of its democracy.

Fears were running high after a political row left the Central Election Commission short-staffed and unable to certify the result.

A shooting during the election in the northwestern Lac region, in which an opposition activist was killed and a Democrat candidate wounded, strained nerves further.

The police said they had not yet arrested anyone.

Rama lost the last parliamentary election in 2009 and four people were shot dead by security forces when opposition protesters took to the streets.

"Important that all parties in Albania fully respect the result of their parliamentary election today," Swedish Foreign Minister Carl Bildt said on Twitter. (Writing by Matt Robinson, editing by Gareth Jones)