Monsignor Leo Cushley, who was yesterday unveiled as Cardinal O'Brien's successor following the enforced resignation, said he would "work hard to get this business sorted out", adding it was crucial any investigation had to be open and transparent.
Mgr Cushley, 52, is currently head of the English-language section of the Vatican's Secretariat of State and has been a close collaborator of both Pope Benedict XVI and Pope Francis.
Cardinal O'Brien resigned as Archbishop of St Andrews and Edinburgh five months ago after admitting decades of sexual behaviour with other clerics and was exiled by the Vatican from Scotland in May. He remains a cardinal.
Asked of his intentions to investigate the scandal, Mgr Cushley said: "If there is an investigation it won't be up to me. It will depend on the Holy See.
"But I will be at Pope Francis's disposal and can assure you that any allegations will be properly investigated.
"I will work hard to get this business sorted out. It's very important it is an open and transparent process. We need to do this truthfully. We need truth and honesty to start a fresh chapter."
He said he had very little knowledge about Cardinal O'Brien's situation other than what he had read in the press.
He added that he had never discussed it with Pope Francis, despite working closely with both him and Benedict in Rome.
He added: "I've never discussed this issue with Pope Francis but be assured he was informed in detail of what was going on here.
"In preparation (for his appointment) there would be a complete profile of the Archdiocese and this situation."
Mgr Cushley said the priests and people within the Archdiocese were anxious to move on from the scandal but denied the Church was at a historic low ebb.
He said: "I think (the Church) has taken a bit of battering. I think that is fair. But also, as I said before, I think the fundamentals are good and they are right.
"The priests and the people are very anxious to move on, and I am with them on that."
The new Archbishop-elect will take over the governance of the Archdiocese from Archbishop Philip Tartaglia, who has been Apostolic Administrator since Cardinal O'Brien quit.
Mgr Cushley, who was born in Airdrie, North Lanarkshire, will be ordained as Archbishop in St Mary's Cathedral, Edinburgh, on September 21.
He expressed surprise at being appointed Archbishop, given his background as part of the Vatican's diplomatic team for the past two decades and was frank about his lack of pastoral work, having spent just six years as a parish priest in the Diocese of Motherwell.
Asked, given the unique circumstances within which he takes over the Archdiocese, where it was "congratulations or commiserations", he replied: "My mother is delighted. I'm no longer miles and miles away.
"I think I'm happy to be back. There's challenges but I'm looking forward to it."
He also described the challenges of his new role as comparatively easy compared with previous situations he has faced.
He referred to referred to his experiences working in Burundi towards the end of the civil war, during which 400,000 people, including a Vatican ambassador working with Mgr Cushley, were killed.
He said: "That gives you an idea of what we are talking about. You have to get a grip of yourself and say 'What do I believe here?' because your life is on the line.
"So you have a very different attitude to coming to this kind of thing. It is a question of getting to know this country ... getting to know the archdiocese, the priests and the people, and taking it from there. It is a comparatively easy task.
"I think there will always be a difference between what the Catholic Church preaches and teaches and the individuals who propose that message, who are sinful."
Mgr Cushley also said, that despite allegations of hypocrisy following the Cardinal O'Brien revelations, he would not shy away from involving himself in political issues and also reaffirmed the Catholic Church's opposition to same-sex marriage.