Unofficial returns released by election officials, with 99 per cent of the vote counted, showed the Socialists with at least 204 seats in the 386-seat parliament.
The liberal Alliance of Free Democrats was second with 70 seats and the governing conservative party, the Hungarian Democratic Forum, had only 37. The remainder were spread among smaller parties.
Although victory for the Socialists was widely predicted, its scale came as a shock. Even though the party can now rule alone, it will undoubtedly seek to form a coalition with the AFD, with which it shares many goals.
In the run-up to yesterday's concluding round of voting, HSP leaders made it clear that they did not want a return to one-party rule.
'A coalition is a political necessity,' Laszlo Bekesi, a former and perhaps future finance minister, said. 'In other words, this is one of the important conditions for effective government, not just a parliamentary mathematical necessity.' One sticking point in coalition negotiations will almost certainly be the question of the premiership.
Gyula Horn, the HSP leader and foreign minister in the reform Communist administration of 1989-1990, is said to be keen on the job, while the AFD would staunchly oppose his nomination. Mr Horn served in the Communist militias which suppressed the 1956 uprising.
In the 1980s he turned against Communism and was a prominent reformer in the party's internal struggles. In 1989 he made the decision to allow thousands of East German refugees into the West, creating an exodus which led to the collapse of East Germany and the fall of the Berlin Wall.
Balint Magyar, the campaign manager for the AFD, has said that the Free Democrats would feel 'exposed and vulnerable' joining a government in which the HSP could impose its will.
Whatever the final composition of Hungary's next government, leaders of both the Socialists and the Free Democrats have warned that it will be at least two years before they can achieve a marked improvement in general living standards.
The perceived worsening of the economic situation - unemployment is now at 12 per cent while inflation is running at about 22 per cent - was one of the main reasons for the crushing defeat delivered at the polls to the HDF.
Along with other centre-right parties elected to power in Eastern Europe in the first flush of freedom, following the collapse of Communism in 1989, they presided over a period of unprecedented economic transition. In addition to many new winners, many more people ended up feeling they had lost out.
As with their counterparts in Lithuania and Poland, the Hungarian conservative government was blamed for many of the economic difficulties that swept the country after 1989. They were perceived to be arrogant and out of touch with ordinary people.
Peter Boross, the outgoing HDF Prime Minister, said yesterday that his party will form an active opposition to the next government. 'We will see what the opinions of voters are in two or four years' time,' he told reporters rather sourly as he cast his vote.
Representatives of the HSP, meanwhile, sought to provide reassurances that their victory will not mean a return to the Communist dark ages.
'The West has nothing to fear from a Socialist or Socialist-led government,' Laszlo Kovacs, the HSP's likely candidate for foreign minister, said. Socialist leaders have sought to reassure foreign investors, who have pumped dollars 7bn (pounds 4.6bn) into Hungary since 1989.