Twersky was born in Boston in 1930, and attended Boston Latin School and Hebrew College, which was then known as "Hebrew Teachers' College". Much of his Torah knowledge was acquired through private study under the tutelage of his father Meshullam Zushe Twersky (his predecessor as the Talner Rebbe of Boston) and his future father-in-law Joseph B. Soloveitchik, rather than through formal yeshiva instruction . He then graduated from Harvard in 1952, where he majored in history. In 1949, he spent a year at the Hebrew University of Jerusalem.