Hurwitz was born in 1919 in east London to parents of Russian-Jewish ancestry. He began studying the violin at age five with a local teacher and won a scholarship to the Royal Academy of Music at 14. There he studied with Sydney Robjohns, which was, writes The Guardian, his first introduction to the Viennese-Hungarian style of playing.
During World War II, Hurwitz joined the Royal Army Medical Corps and played in its band. In 1943 he was sent to the Middle East and returned to London in 1944 to spend the last part of the war playing in a group called Stars in Battledress.
After the war, the violinist founded the Hurwitz Quartet, whose members also joined the English Chamber Orchestra when it was established in 1948. Hurwitz also led the Melos Ensemble from 1956-72 and made many recordings with the group, which often collaborated with Benjamin Britten. According to The Guardian, Britten regularly consulted Hurwitz about technical aspects of violin writing.
In 1969, Hurwitz became first violinist of the New Philharmonia Orchestra, leaving it in 1971 after having joined the Aeolian String Quartet as first violinist the previous year. He remained with the ensemble until it disbanded in 1981.
Hurwitz was also a professor at the Royal Academy of Music for many years, then taught privately at his home. In addition, he was an influential figure at London's instrument auctions, particularly with respect to the purchase of bows. Hurwitz himself played a 1603 Amati instrument.
The Guardian writes that "It was as a leader that Hurwitz found his playing and personal strengths most happily employed, his quiet tact and good humour always calming any troubled waters. Never one for losing his temper or throwing his weight about, and invariably friendly to those around him, he would always see the funny side of some impossible situation — often involving the notorious egos of conductors and soloists — and was able to dig his fellow musicians out of whatever hole they were in, often with large lashings of his incredible humour."