Fidesz chief Viktor Orbán conceded his election defeat in a phone call to Gyurcsány, Socialist Chairman István Hiller said before claiming victory for his party.
Voter turnout was 60.3%, compared with 67.8% in the April 9 first round, the National Election Bureau (OVI) said. The Socialists took 190 seats and their coalition partner Alliance of Free Democrats won 20; Fidesz won 164 seats and the smaller opposition party Hungarian Democratic Forum took 11, with one independent candidate winning a single district.