Isaac was the sixth son of Robert Bargrave, of Bridge, Kent, and was educated at Clare Hall, Cambridge, where he graduated B.A.

In 1612 he held the office of 'taxor' at Cambridge, and he acted in the Latin comedy Ignoramus performed at the university before James I on 8 March 1615 and written by George Ruggle of his college.

Shortly afterwards Bargrave went to Venice as chaplain to Sir Henry Wotton, the English ambassador there, and befriended Paolo Sarpi.

In the last years of James I's reign Bargrave had preached a sermon which threw him into disfavour with the court; but as dean of Canterbury he supported the policy of Charles I.

Bargrave claimed precedence over the deans of London and Westminster, and engaged in a dispute with William Somner, the registrar of the diocese.

When the bill for the abolition of deans and chapters was introduced by Sir Edward Dering, the first cousin of his wife, he was fined £1,000 as a prominent member of convocation.

At the beginning of the First English Civil War, in August 1642, Edwin Sandys, a parliamentary colonel who had been on good terms with Bargrave, occupied the deanery.