The two-thirds supermajority that Magyar's conservative Tisza party secured in Sunday's election means it will have a free hand to strengthen the rule of law and potentially unlock billions in European Union funding but analysts say it will have to really deliver on reforms to reap the benefits.
Outgoing nationalist Prime Minister Viktor Orban, in power for 16 years, had turned Hungary from the EU mainstream while maintaining warm ties with Russia despite the war in Ukraine.
He repeatedly clashed with the bloc over the rule of law and human rights, resulting in the freezing of billions of euros in funds.
Magyar told a news conference his government would have many pressing tasks, including amending the constitution to limit the number of terms someone can serve as prime minister to two.
"We will do everything to restore the rule of law, plural democracy and the system of checks and balances," he said.
Magyar said the constitutional amendment would apply to Orban, meaning he cannot become prime minister again.
"He had a great opportunity to do huge things in the national interest to ensure that Hungary became a developing European country ... He did not use this chance, but abused it," he said.
Critics of Orban say his time in office resulted in economic stagnation, international isolation and oligarchs amassing wealth.
His supporters say Orban - who secured public endorsements from US President Donald Trump, Russia and populist-nationalist leaders in Europe - defended Hungary's sovereignty and traditional values.
Magyar said the election result showed that Hungary had decided to "change the regime" and choose a pro-European path.
"Hungarian people yesterday, exactly 23 years after the referendum about our EU membership, confirmed Hungary's place in Europe," he said.
The Tisza leader called on President Tamas Sulyok, who is backed by Orban's Fidesz party, to ensure that the transfer of power happened as soon as possible.
He also reiterated his demand that Sulyok resign.
Hungary's president, a largely ceremonial figure, must convene a new parliament within 30 days of an election.
MPs then elect a new prime minister.
In Budapest many residents were recovering from wild celebrations that had the streets of the capital thronged with revellers well into the early hours.
Among those letting themselves go was Zsolt Hegedus, a leading candidate for health minister, whose dance moves on stage following Magyar's speech went viral online.
Tisza supporters spoke on Monday morning of their optimism about the end of Orban's adversarial relations with the EU.
"We are part of the European Union and we want to stay within the European Union," network engineer Gyula Ferenc Teleki said.
Meanwhile, the Hungarian forint surged almost 3 per cent to a more than four-year high against the euro while the Budapest stock exchange gained 3.5 per cent in anticipation of EU funds starting to flow.
EU leaders, Ukraine's President Volodymyr Zelenskiy and Trump's Democrat opponents in the United States have all warmly welcomed Magyar's victory.
Orban has blocked a 90-billion-euro ($A149 billion) loan to war-battered Ukraine.
German Chancellor Friedrich Merz said that decision-making in the EU, including with regard to Russia, would become easier after Magyar's win.
The Kremlin said Russia respected the choice of Hungarian voters and would continue what it called "pragmatic ties" with the new government in Budapest.
However, Russian hardliners cast Orban's defeat as a serious blow that would unlock EU military funds for Ukraine to fight on against Russia for another year.
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