In 1844, Westcott entered Trinity College, Cambridge, where he was invited to join the Cambridge Apostles. He became a scholar in 1846, won a Browne medal for a Greek ode in 1846 and 1847, and the Members' Prize for a Latin essay in 1847 and 1849. He took his BA degree in January 1848, obtaining double-first honours. In mathematics, he was twenty-fourth wrangler, Isaac Todhunter being senior. In classics, he was senior, being bracketed with Charles Broderick Scott, afterwards headmaster of Westminster School.