Former Bihar SC/ST welfare minister Jitan Ram Manjhi became the state's new chief minister on Tuesday.
The Janata Dal (United) had announced Manjhi's name for the chief ministerial post in the state on Monday.
"All I have to say is the issue that we had taken up has been successful," said Janata Dal (United) President Sharad Yadav.
Kumar said that he had taken the decision to resign, as it was his moral responsibility to own up for the party's poor performance in the recently concluded general elections.
Kumar had stepped down from the top post on Saturday, but did not recommend dissolution of the state assembly.
The JD-U won only two parliamentary seats out of 40 in Bihar while Bharatiya Janata Party (BJP)-led National Democratic Alliance (NDA) cornered 31 seats.
Modi's JP won 282 seats, 10 more than the majority required to rule.