With about 95 percent of the vote counted, Tisza was set to win 138 seats in the 199-member parliament, enough for a two-thirds constitutional majority, data from the National Election Office (NVI) showed.
Orban’s ruling Fidesz party was expected to win 54 seats.
Voter turnout was a record 77.8 percent, according to provisional data, in an election widely seen as crucial for Hungary and Europe.
After 16 years in power, Orban congratulated Tisza leader Peter Magyar, who is expected to form a new government.
"The election results are not final yet, but the situation is understandable and clear," Orban told supporters at Fidesz campaign headquarters in Budapest.
"The result is painful for us, but clear," he added.
Observers said his defeat was not only a watershed moment for domestic politics but could also send shock waves through right-wing political circles in the European Union and beyond.
Sunday's vote was closely watched in Warsaw, with Magyar pledging to strengthen ties with Poland if he takes power.
Polish Prime Minister Donald Tusk said ahead of the vote that the Hungarian election could mark a turning point for Europe following a recent wave of gains by pro-Russian and Eurosceptic groups.
'East or West'
Magyar had cast the election as decisive for Hungary’s future, framing it as a choice between "East or West, a failing state or a functioning one, corruption or a fair public sphere."
He said a new government would prioritise anti-corruption measures, strengthen Hungary’s position in the EU and NATO, unlock frozen EU funds and improve relations with Poland.
Analysts say the end of Orban's rule could have significant implications for Ukraine, potentially easing Hungary’s opposition within the EU and opening the way for new financial support for Kyiv.
European Commission head Ursula von der Leyen said after partial results that "Hungary has chosen Europe."
Orban’s departure would also deprive Russian President Vladimir Putin of a key ally within the EU, the Reuters news agency reported.
Despite support from US President Donald Trump and other right-wing figures, it said Orban’s campaign was weakened by voter dissatisfaction over economic stagnation, rising living costs and allegations that business figures close to the government had accumulated significant wealth.
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Source: TVP Info, Reuters, IAR, PAP