With only a fraction of official results declared it was impossible to forecast the final outcome, but Mr Nano told reporters that his party's data showed it had won 60 of 115 seats contested under majority voting rules and together with other leftist parties would hold a two-thirds majority in parliament.
"A centre-left coalition will control two-thirds of the new parliament for sure," Mr Nano said, adding that the Socialists stood to win half of 40 seats being contested under proportional rules.
Mr Nano, who was released from prison only three months ago, suggested that the two-thirds majority he expected - which would be sufficient to outvote or even remove Mr Berisha - meant the president should quit. "With those figures that are already confirmed and the support that the Albanian people gave the centre-left coalition, this problem is already resolved," he said.
Earlier, Mr Nano had claimed that there were "a lot of rats leaving the sinking ship". He added: "The only thing Berisha has left is his charisma, but I am more charismatic than he is. I have the president to thank for that, because he put me in prison for four years."
The official count has given the Socialists a handful of seats in the south, heartland of months of rebellion against Mr Berisha. The president's Democratic Party, also quoting its own data, said it had won 17 seats.
Earlier in the day as Albania, torn apart by months of political turmoil, economic collapse and gang violence, finally held its much-anticipated emergency elections, the whole country was in a nervous state of limbo. Barely a car ventured out on to the roads.
Bars and squares were eerily silent. The beaches were deserted, partly because of fears of snipers shooting from smugglers' boats. Even the police, which was supposed to be deployed at maximum strength across the country, abandoned checkpoints and vanished entirely from pollings stations and city streets.
Despite fears of mass violence, the election itself passed off in most places in an anxious hush, with only sporadic reports of shootings and armed attacks on voters. In towns like Lushnja, which saw rabid anti-government rioting earlier this year and a shoot-out at a presidential rally earlier this week, there were whispers in the few bars to stay open that the ruling Democratic Party was intimidating voters and that the result, in some polling stations, had been decided in advance.
One man died in a hand-grenade explosion, and there were bursts of Kalashnikov fire throughout the afternoon. "Everything is fine, absolutely fine," said one returning officer with suspicious insistence in his polling station, a disused funeral parlour.
In another polling station, seals were missing from ballot boxes and officials were overheard boasting about how much of the vote each party had been apportioned in advance.
In one area of the north-eastern part of the country, the Mat, voting was not possible because ballot papers had not been delivered, while in another, Merdita, voters were kept off the streets by armed gangs controlled by the right-wing Republican Party.
Even in more relaxed towns like Kavaja, not all was well. One of two vote-counting centres had been abandoned by officials of all parties except the Democratic Party, which spoke confidently of the re-election of its candidate, the party chairman Tritan Shehu.
The most normal place was the capital, Tirana, where the only upsets were over minor details of voter registration.