Jadue will assume office for a four-year term on January 15, officially ending the mandate of former president Harold Mayne-Nicholls, a FIFA official who appointed Bielsa in 2007.
Bielsa said before the original election in November that he would not stay on unless Mayne-Nicholls remained in office.
ANFP spokesman Wilfredo Sequeira said Jadue beat rival Ernesto Corona by 27 votes to 21.
Spanish-born Jorge Segovia beat Mayne-Nicholls in the first election in November but the result was declared null and void by a tribunal because of a clash between his personal business affairs and his position as president of first-division club Union Espanola.
Bielsa signed an extension to his contract with the ANFP after the 2010 World Cup finals, Chile's first since 1998, in South Africa where the team reached the second round before losing to Brazil.
Jadue, a 31-year-old lawyer, told a news conference he would ask for a meeting with Bielsa to try to convince him to go back on his decision to quit.
"I ask him to receive me so that he can carry out his commitments. He'll have all the guarantees and the right people so he can work," said Jadue, president of promoted La Calera.
