With 98% of votes counted, Tisza had won 138 seats in the 199-member parliament, while Prime Minister Viktor Orban's Fidesz-KDNP alliance took 55 seats. The far-right Mi Hazank was also projected to enter parliament with six seats.
"We have regained our state, we have freed it", Magyar told thousands of supporters gathered in central Budapest. "TISZA not only won, but will have a two-thirds constitutional majority".
He said the result gave his party a mandate to deliver an "easy and peaceful transformation".
Magyar said Hungary would "again be a strong and reliable ally of the European Union and NATO" and confirmed he would travel first to Warsaw "to rebuild the thousand-year Polish-Hungarian friendship".
"Then I will go to Vienna, and then to Brussels, to bring back the European Union funds that belong to Hungary", he said.
Addressing chants of "Russians go home" from the crowd, Magyar replied: "We will be a country that serves no one".
He also called on the president and the heads of public institutions appointed under Orban's parliamentary majority to step down voluntarily.
"Otherwise, we will dismiss you ourselves. This regime is over", Magyar said.
Magyar said Orban had called to congratulate him on the election victory and that he had told the prime minister they now shared responsibility for restoring unity in Hungary.
"Hungarians said 'yes' to Europe and to a free Hungary", Magyar said.
(jh)
Source: PAP