Justin Welby, the Archbishop of Canterbury, said he "consistently put the interests of others ahead of his own and, in so doing, provided an outstanding example of Christian service".
The palace added: "The Royal Family join with people around the world in mourning his loss."
The flag at Buckingham Palace was lowered to half-mast and a notice was posted on the gates following the announcement of the duke's death.
Bunches of daffodils, tulips, roses and lilies were among floral tributes placed by people outside the palace, while crowds began to gather at Windsor Castle.
The BBC's royal correspondent Nicholas Witchell said it was "a moment of real national sadness" and "a moment of sadness, most particularly of course, for the Queen losing her husband of 73 years - a bigger span of years than most of us can imagine".
He said Prince Philip had made "a huge contribution to the success of the Queen's reign", describing the duke as "utterly loyal in his belief in the importance of the role that the Queen was fulfilling - and in his duty to support her".
"It was the importance of the solidity of that relationship, of their marriage, that was so crucial to the success of her reign," he added.
A bank of photographers and cameramen were lined up around the growing number of tributes at Buckingham Palace on Friday afternoon, according to BBC News reporter Marie Jackson.
Rhea Varma, from Pimlico, pulled up to the gates on her bike to lay flowers and a note saying Rest in Peace Duke.
She said: "It's just super sad. I think it might be the beginning of big change in our country. Without him, the Queen might not reign much longer."
To her, the duke was "the kind of stability that's so old-fashioned it's difficult to comprehend. He was a rock who brought integrity."
The prince married Princess Elizabeth in 1947, five years before she became Queen.
Tributes are paid from around the world to the Queen's "beloved" husband of 73 years, the longest-serving consort in British history.
www.bbc.co.uk