Jobbik came second in 18 out of 19 counties, ahead of the Socialists, as in the May European elections.
It won control of 14 towns and villages. The party has been accused of anti-Semitic and anti-Roma rhetoric.
Fidesz won all but six districts of the capital Budapest, where its mayoral candidate Istvan Tarlos was re-elected.
It has a two-thirds majority in the national parliament and the local election consolidated its grip over the country's politics. It came top in nearly all the big towns and cities.
Jobbik is the third-largest party in Hungary's parliament and has three MEPs. It has tried to soften its image recently, but remains committed to defending conservative Hungarian values and thwarting integration of Roma communities, many of whom are desperately poor and marginalised.
The BBC's Nick Thorpe in Budapest writes:
"From 2018, Jobbik will govern this country," Jobbik leader Gabor Vona told cheering supporters in Budapest as the election results came in.
But he faces an uphill task to overtake Fidesz. His main problem is that Prime Minister Orban has stolen most of his Eurosceptic clothes.
Jobbik did well in this election, but not as well as it hoped. In 18 of 19 counties, it came second to Fidesz, proving that it now has nationwide support.
In the countryside, Jobbik won five towns and nine villages, up from one town and three villages in 2010. Most spectacularly, it won narrowly in the northern industrial town of Ozd, which has a large Roma population.
Jobbik's initial success was largely based on its ability to tap into widespread fear of the Roma in the Hungarian countryside.
In all three  elections this year - parliamentary, European and municipal - it has considerably toned down its rhetoric. It is now trying to appeal to middle-class voters who have abandoned Fidesz, as their standard of living stagnates and their children leave to work abroad.

About 150 workers had to leave the building on Tuesday morning as the fire took hold. No injuries are reported.
The hotel - which is famously where Princess Diana spent her last evening - is currently closed for renovations and there are no guests staying there.
The operation will take "a long time" as the fire is difficult to access, a fire service spokesman said.
The alert was raised at 07:00 local time (06:00 GMT) and about 60 firefighters and 15 fire engines were sent to the scene.
The blaze started on the seventh floor, took hold in the attic and spread to the roof.
Paris police tweeted that there was a "major fire", and advised motorists to avoid the Place Vendome area.
The area was blocked off at the height of the morning rush hour, while firefighters on cranes sprayed the flames.
Fire captain Yvon Bot said the fire was unusually difficult to access and control.
"There is no longer any risk of it spreading, but the operation will still take a very long time," he said.
"We are having to go metre by metre from the roof. It is too small to access from the interior."
He said a "large" section of the building was affected.
The hotel is a historic building in central Paris.
Princess Diana dined there on 30 August 1997. Later that night she and her companion Dodi al-Fayed were killed in a car crash. He was the son of the hotel's owner, Egyptian billionaire and Harrods owner Mohamed al-Fayed.
It was also the hotel of choice for Charlie Chaplin, Coco Chanel and Ernest Hemingway.
The Ritz was due to re-open in March after a three-year closure for multi-million euro renovations.
The Ritz name, used for hotels around the world, comes from a Swiss hotelier called Cesar Ritz, who set up the Paris Ritz and the London Ritz more than a century ago.
Both hotels are now independently owned. The Barclay brothers - newspaper owners Sir Frederick and Sir David Barclay - bought the London Ritz in 1995.
The far-right Jobbik party in Hungary has made gains in local elections won convincingly by Prime Minister Viktor Orban's ruling nationalist Fidesz.
[NEXT_CONCEPT]
A fire on the top floor of the five-star Ritz hotel in Paris has been brought under control.