The 62-year-old had served as bishop of the diocese since 2004.
The Archbishop of Wales, Dr Barry Morgan, paid tribute to the courage and faith shown throughout his ministry.
"In the four short years Bishop Tony spent in Bangor he endeared himself to the diocese and made his mark on it. He was a feisty and forthright man, not afraid to express his vision, yet had a warm pastoral heart and would go out of his way to help anyone," he said.
"He faced his last illness with resilience, faith and hope. His tremendous courage and faith strengthened and inspired others.
"He showed great ministry in his weakness, continuing his work in hospital - even taking confirmations - while never afraid of showing his weakness and vulnerability. He will be hugely missed."
Anthony Crockett was elected 80th Bishop of Bangor in 2004. Previously he was Archdeacon of Carmarthen and incumbent of three rural churches in West Wales.
The Dean of Llandaff, the Very Rev John Lewis, also paid tribute to Bishop Crockett.
He said, "Tony was a great friend and splendid company. His life and ministry were a true pilgrim's journey and he allowed many to share that journey with him. He was perceptive and honest about the present and inspirational and supportive for the future. We shall miss him."
A Welsh speaker and native of Pontypridd, he graduated in Classics and in Theology at King's College, London, and completed his pre-ordination training at St Michael's College, Llandaff. He was ordained deacon in 1971 and priest in 1972, and served curacies in Aberdare and Whitchurch in the Diocese of Llandaff, before becoming a rural parish priest in Cardiganshire from 1978-1986.
During this time, he became an Examining Chaplain to the Bishop of St Davids and Secretary of the Provincial Selection Board. From 1986-1991, he was Rector of Dowlais, and from 1991-1999 Secretary of the Board of Ministry of the Church in Wales. He became Archdeacon of Carmarthen in 1999.
Philip Price QC, chairman of the Standing Committee of the Governing Body of the Church in Wales, said, "Bishop Tony's death is a sad loss to the religious life of the nation. He thought deeply about the ministry of the church and all people and expressed himself clearly and firmly in his preaching and his writing.
"His scholarship was allied to a real pastoral understanding and he worked tirelessly for the diocese and the province. His courage in this long illness is an inspiration and I personally marvelled at his ability to keep his concern for others at the forefront of his mind throughout."
Bishop Crockett had a special interest in pilgrimage, having walked 1,000 miles from Le Puy to Santiago de Compostela on sabbatical in 1995. He recently instigated the third-ever official pilgrimage to Bardsey, an island off the Lleyn peninsular, North Wales, which took place on June 21, although he was too ill to join in.
He was also a passionate supporter of the ordination of women as both priests and bishops.
He leaves his wife, Rev Caroline Owen, and three children, Kate, Thomas and Elizabeth.