Rt Rev Eagles, 57, has been Archdeacon to the Army since 2011 and Deputy Chaplain General of the Royal Army Chaplains' Department since 2014.
His appointment comes after the resignation of his predecessor the Right Rev Robert Mar Erskine Paterson in November, 2016.
Commenting on his appointment he said: "I am both thrilled and deeply honoured to be coming to The Diocese of Sodor & Man. My hope is that I may support the people of the Isle of Man in their spiritual and historic identity and in looking confidently to the future.
"Among many other tasks, the bishop is to be chief pastor, a focus of unity, and a leader of mission. I therefore hope to encourage all Christians on the island, to support everyone who lives here, and to lead the Church in its work of caring for people in every community."
Rt Rev Eagles, who trained at St Stephen's House, Oxford after completing a degree at King's College, London has been ordained for 27 years.
After serving as a curate in Middlesex he became an Army Chaplain in 1992. In 2013 he was appointed as an Honorary Chaplain to the Queen and in 2015 was appointed Honorary Canon of Salisbury Cathedral.
His appointment has been welcomed by many prominent figures from the Church of England.
The Archbishop of York, Dr John Sentamu said: "Peter brings a wealth of experience in parish ministry and in army chaplaincy, and has many spiritual gifts to help him lead the church forward in the mission of God in the diocese to which he has been called.
"My prayers are with Peter and his family as they prepare to move to the Isle of Man. May they know the grace and power of the Holy Spirit strengthening them for all that lies ahead."
The Right Revd Nigel Stock, Bishop at Lambeth and Bishop to the Forces, said: "Peter Eagles has been a superb colleague over the time that I have been Bishop to the Forces.
"Always considerate he has been a great support to his fellow chaplains as Archdeacon to the Army.
"His learning is worn lightly but is very impressive. He is a man of prayer with a pastoral heart, but unafraid to face challenges. I know that he will be a highly effective bishop and that the people of Sodor and Man will be very well served."