But in order to make constitutional changes, the party would need to garner a two-thirds majority in the 386-seat parliament. This remains to be seen after a second round vote to be held in two weeks time on April 25.
Fidesz's leader, Viktor Orban, served as prime minister between 1998 and 2002.
The outgoing Socialists, who have had a relatively long run in power for eight years, won almost 20 per cent of votes. The party's leader, Ildiko Lendai, has so far refused to admit defeat.
Close on the Socialists' heels was the far-right Jobbik party, with almost 17 per cent of votes. It already has seats in the European Parliament.
More of a surprise in these elections in Hungary, was the performance of the new left-wing green party, the LMP. It also won its first seats in parliament with 7.42 per cent of votes.
Turnout was slightly lower than expected at 64.29 per cent, down three points from 67.83 per cent in the first round of general elections in 2006.
The economic background to the polls may have influenced the vote. Hungary was one of the countries worst hit by the global economic crisis in 2008.
It escaped bankruptcy thanks to a 20-billion-euro bailout from the IMF, the World Bank and the European Union.
Eighteen months later, the country's economy is healthier and that's thanks to the Socialist government's rigorous budgetary programme. But tax hikes and wage and pension cut rarely help the popularity polls.