After graduation from Columbia University, he studied at the General Theological Seminary of the Episcopal Church in the United States before ordination as deacon on May 17, 1822 and priest on November 13, 1825 by Bishop John Henry Hobart of the Diocese of New York.

On December 29, 1842, he was consecrated as assistant Bishop of Massachusetts; he served as diocesan bishop on the death of Alexander Viets Griswold from 1843 until his death.

Eastburn attended the first Lambeth Conference in 1867 and was associated with the evangelical school of Episcopalian churchmanship.

His tenure as diocesan bishop was marked by considerable conflict over Tractarianism both locally and nationally, particularly at the Church of the Advent in Boston.

He is buried at St. Paul's Church, in Dedham, Massachusetts.