The vote Sunday was being closely watched across Europe and in the United States, with US President Donald Trump endorsing Orban, a self-described "thorn" in the EU's side and a defender of "illiberal democracy" who is also close to Russian President Vladimir Putin.
"The election results, though not yet final, are clear and understandable; for us, they are painful but unambiguous," Orban, 62, told reporters.
"We have not been entrusted with the responsibility and opportunity to govern. I congratulated the winning party," he said.
Magyar, 45, a political newcomer and former government insider, posed on social media that Orban "just called to congratulate us on our victory".
Turnout in the election reached a record 77.8 percent at 6:30 pm (1630 GMT).
'Full of hope'
"I'm really excited. I came to this event full of hope... I'm optimistic about the change," Orsolya Rozgonyi, a 28-year-old HR manager, told AFP amid Tisza supporters who gathered in Budapest in a festive atmosphere.
The atmosphere was more subdued among hundreds of Fidesz supporters who gathered near the press centre, cordoned off from the press.
Orban, who was seeking a fifth straight term, has transformed his country into a model of "illiberal democracy", publicly clashing with Brussels over rule-of-law issues, as well as over support for war-torn Ukraine.
Magyar burst onto the scene just two years ago, promising to fight corruption and offering better public services, amassing support against a backdrop of economic stagnation, despite an electoral system skewed in favour of Orban's Fidesz party.
Both camps have alleged foreign interference during the campaign in the central European country of 9.5 million people.
US Vice President JD Vance visited Hungary this week to rally with Orban, attacking the alleged interference in Hungary of Brussels "bureaucrats", while Trump has promised to bring US "economic might" to Hungary if Orban's party secures victory.
'No infringement'
After casting his ballot, Orban repeated warnings of a "major crisis" awaiting Europe.
"Fortunately we have a lot of friends in the world. From America to China to Russia and the Turkish world," he said.
Orban has focused on making Ukraine the central topic of his campaign, portraying the neighbouring country, which is fighting off a Russian invasion, as "hostile" to Hungary.
He also vowed to continue his crackdown against "fake civil society organisations, bought journalists, judges (and) politicians".
Maria Toth, a 31-year-old stay-at-home mother of two, told AFP at a Budapest polling station that "it is so important for us that Viktor Orban stays in power".
"I feel Hungary is under siege from so many directions and big powers like Brussels are trying to dictate how we live," she added.
Orban's chief of staff Gergley Gulyas said earlier Sunday that he didn't see, for now, "any infringement that would put the final result in question".
After voting, Magyar called on voters to report suspected vote-buying, intimidation or other violations, while urging calm.
