Prime Minister Viktor Orban is seeking his third consecutive term and fourth overall. He has transformed Fidesz from a liberal party formed in the 1980s to a right-wing populist outfit, which has campaigned this election on an anti-immigration platform.
Opinion polls before the vote had consistently put Orban and Fidesz party 20 or more points clear of their nearest rivals, Jobbik, a far-right party that has been moving towards the centre, and the centre-left Socialists.
The National Election Office initially said it did not intend to disclose any results while voters were still waiting outside polling stations.
Orban claims that the opposition - collaborating with the United Nations, the European Union and wealthy philanthropist George Soros - wants to turn Hungary into an "immigrant country", threatening its security and Christian identity. "Safety is first", said Julia Scharle, 27, holding her child outside the voting district where Orban cast his vote.
Hungary's National Election Office reported that over 5.3 million voters of the 8.3 million eligible voters had cast ballots by 6:30 p.m. (1630 GMT), fir a turnout of 68.1 percent.More news: San Diego gun store employee talks about sale to YouTube shooter
"High turnout means, most probably, less mandates for Fidesz than in the previous term", said Peter Kreko, director of think tank Political Capital.
"The government serves the interest of the country, because we love our country and we are fighting for the future of our country".
"Theoretically everything is still possible as we don't know the data yet. but in Hungary a two-thirds victory is possible if neither side loses more than 10 districts and there is a difference of at least 20 percent between the victor and the runner-up", Gulyas said. Vona said the question was not about migration into Hungary but about the large number of Hungarians who were leaving the country and heading to Western Europe in search of higher wages and better prospects.
The EU has struggled to respond as Orban's government has, in the view of its critics, used its two landslide victories in 2010 and 2014 to erode democratic checks and balances.
On Friday, at his closing campaign rally, Orban vowed to protect his nation from Muslim migrants, saying: "Migration is like rust that slowly but surely would consume Hungary".More news: Welbeck's brace helps Arsenal overcome Southampton 3-2 at Emirate
However, the polls could be unreliable as one-third of voters are undecided and many hide their voting preference.
In March the government gave pre-election handouts to millions of families and pensioners.
Orban is expected to continue his economic policies, with income tax cuts and incentives to boost growth, analysts have said. The Socialists came in third with 14 percent.
If borne out, Mr Orban will likely seize on the results as vindication of his clashes with European Union institutions over his hardline anti-immigration policies and rejection of the EU's refugee resettlement programme, as well as his moves to clamp down on civil society groups.
Fidesz may even be on track to win its coveted two-thirds "super-majority", which would grant it wide powers to press ahead with controversial measures and change the constitution.More news: US, China step up trade war