"It's onward and upward," Mr Bradley told BBC Radio Merseyside.
"I secured well over 50% of the vote last night and I am very proud this morning to be the leader of Liverpool City Council."
The challenge was launched by councillors who felt the party needed a change of direction, and Mr Bradley said that change was on his agenda.
"I think any political party, or any business, has to redirect itself and I think after 10 years it is time for a redirection."
Mr Bradley said he would be selecting a new cabinet next week, and promised to bring new people in to "re-energise" and take the city forward.
He added: "There's lots of other challenges in this city and it would be inappropriate for me to get caught up in what has happened."