He has been the Bishop of Durham for little over a year and succeeds Dr Rowan Williams, who is stepping down at the end of the year.
Speaking to the press today, Bishop Welby said his appointment as Archbishop of Canterbury was "exciting" and "never expected".
"It is exciting because we are at one of those rare points where the tide of events is turning, and the church nationally, including the Church of England has great opportunities to match its very great but often hidden strengths," he said.
Despite the challenges of falling attendance and divisions over sexuality and women bishops, Bishop Welby said he was "utterly optimistic" about the future of the Church of England.
He was also unequivocal about his support for women bishops.
"This is a time for optimism and faith in the Church. I know we are facing very hard issues. In 10 days or so the General Synod will vote on the ordination of women as bishops.
"I will be voting in favour, and join my voice to many others in urging the Synod to go forward with this change."
Acknowledging the deep differences over sexuality, Bishop Welby said he would "listen very attentively to the LGBT communities".
"It is absolutely right for the state to define the rights and status of people co-habiting in different forms of relationships, including civil partnerships," he said.
"We must have no truck with any form of homophobia, in any part of the Church.
"The Church of England is part of the worldwide church, with all the responsibilities that come from those links.
"What the church does here deeply affects the already greatly suffering churches in places like northern Nigeria.
"I support the House of Bishop's statement in the summer in answer to the government's consultation on same sex marriage.
"I know I need to listen very attentively to the LGBT communities, and examine my own thinking prayerfully and carefully.
"I am always averse to the language of exclusion."
Bishop Welby was educated at Eton and Cambridge, and later spent 11 years working in the oil industry.
He studied for ordination at St John's College Durham and previously served as Dean of Liverpool prior to becoming Bishop of Durham last year.
Paying tribute to his predecessor, he described Dr Williams as a man of "infectious spirituality" and "immense personal moral and physical courage".
"I want to say at once that one of the biggest challenges is to follow a man who I believe will be recognised as one of the greatest Archbishops of Canterbury, Rowan Williams," he said.
Prime Minister David Cameron welcomed the appointment of Bishop Welby as the new Archbishop of Canterbury.
"The Church of England plays an important role in our society, not just as the established church, but in the provision of education, help for the deprived and in furthering social justice," he said.
"I look forward to working with the Archbishop in all of these areas and I wish him success in his new role."
Bishop Welby was congratulated on his appointment by the Methodist Church.
In a joint statement, President of the Methodist Conference the Reverend Dr Mark Wakelin, Vice-President of the Methodist Conference Michael King, and General Secretary of the Methodist Church the Reverend Dr Martyn Atkins said Bishop Welby had demonstrated himself as a "man of spiritual depth and wisdom".
"He has shown great passion and enthusiasm for working together with other churches and we look forward to working with him in the context of the partnership between our two Churches," they said.
"His generous nature, and commitment to serving his communities that have been so greatly valued in Durham will be a huge blessing to the wider Church.
"We look forward to seeing his leadership guide the Church of England in the coming years and will keep him and his family in our prayers as he takes up this significant global role."
WATCH, the pro-women bishops group, said it was delighted with his appointment.
The Reverend Rachel Weir Chair of WATCH said: "We are delighted to hear that Bishop Justin has been appointed to Canterbury.
"He is a strong supporter of women's ministry and has all the skills and experience the Church could hope for in a new Archbishop. We look forward to working with him in future years."