Cosin was born at Norwich, and was educated at Norwich grammar school and at Caius College where he was scholar and afterwards fellow. On taking orders he was appointed secretary to Bishop Overall of Lichfield, and then domestic chaplain to Richard Neile, Bishop of Durham. In December 1624 he was made a prebendary of Durham, and in the following year archdeacon of the East Riding of Yorkshire.
In 1634 Cosin was appointed master of Peterhouse; and in 1640 he became vice-chancellor of the university. In October of this year he was promoted to the deanery of Peterborough. A few days before his installation the Long Parliament had met; and among the complainants who hastened to appeal to it for redress was the ex-prebendary, Smart. His petition against the new dean was considered; and early in 1641 Cosin was sequestered from his benefices. Articles of impeachment were, two months later, presented against him, but he was dismissed on bail. For sending the university plate to the king, he was deprived of the mastership of Peterhouse (1642). He went to France, preached at Paris, and served as chaplain to some members of the household of the exiled royal family. At the Restoration he returned to England, was reinstated in the mastership, restored to all his benefices, and in a few months raised to the see of Durham (December 1660). At the convocation in 1661 he played a prominent part in the revision of the prayer-book, and endeavoured with some success to bring both prayers and rubrics into better agreement with ancient liturgies. He administered his diocese successfully for eleven years; and used a large share of his revenues to promote the interests of the Church, of schools and of charitable institutions. He died in London.