The Rt Rev June Osborne (pictured above, right) was appointed for the role in south Wales in April 2017, instead of another candidate, Dr Jeffrey John who is openly gay.
Dr John claimed he was ruled out of succeeding Dr Barry Morgan because of his sexuality, a claim the Church of Wales denied.
Speaking at her consecration service at Brecon Cathedral on Saturday, Bishop June told BBC Radio Wales: "I'm sure we will go forward really confidently under my leadership."
"The process is very clear and was followed meticulously. I'm confident - and it has been scrutinised - that the process was followed carefully and constitutionally.
It is believed Dr John was the most popular candidate in an electoral college ballot, however he failed to secure a required two-thirds majority.
After the initial process failed to select a candidate, the Bench of Bishops appointed June Osborne - a friend of Dr Jeffrey John - to become the 72nd Bishop of Llandaff.
The consecration service came days after Dr John claimed the contents of emails sent between two bishops proved they opposed his selection because he is gay.
A spokesman for the Church in Wales responded, telling the BBC: "We understand that the private correspondence of the Church in Wales' bishops concerning the process to elect a Bishop of Llandaff has again been put into the public domain by others.
"We are satisfied that the entire process of electing and then appointing the Bishop of Llandaff was carried out properly and fairly and will not be commenting further."