Some 2,000 guests, among them the Prince of Wales and David Cameron, filled Canterbury Cathedral for the ceremony. Archbishops and bishops from around the Anglican Communion, and leaders of other faiths were also present.
In his sermon, Archbishop Welby spoke of "rootedness in Christ" as the key to a society characterised by justice, security and love.
There was "every possible reason for optimism about the future of Christian faith in our world and in this country", he continued.
The Archbishop acknowledged however that it would take "extraordinary Christ-liberated courage" to tackle the challenges presented by the economy, environment and global poverty.
"Courage is released in a society that is under the authority of God, so that we may become the fully human community of which we all dream," he said.
The service started with the Archbishop delivering the traditional three knocks to the cathedral door with his staff.
He was then asked by a young Anglican, "Who are you and why do you request entry" and "Why have you been sent to us?"
He replied: "I am sent as archbishop to serve you, to proclaim the love of Christ and with you to worship and love him with heart and soul, mind and strength."
It was the first time a woman carried out one of the enthronements. The Archdeacon of Canterbury, the Venerable Sheila Watson installed the Archbishop on the diocesan throne.
There were also African flavours to the ceremony, with Ghanaian dancers and a blessing offered in French by the Archbishop of Burundi after Archbishop Welby had taken his seat in the Chair of St Augustine.
Archbishop Welby has previously been involved in reconciliation work in Africa and often spoken of his affection for the continent.
Pope Francis, who was inaugurated in Rome on Tuesday, sent a message to the Archbishop in which he said he looked forward to meeting him "in the near future" and "to continuing the warm fraternal relations that our predecessors enjoyed".