Roy was born in Ranchi, Bihar Province, India, the son of Narenda Kumar and Rajkumari Roy.
Rustum took a Cambridge School Certificate from Saint Pauls School Darjeeling. Rustum studied physical chemistry at Patna University, gaining his bachelors degree in 1942 and master's degree in 1944. The following year he began study at Pennsylvania State University and earned his Ph.D. in engineering ceramics in 1948.
Rustum Roy married fellow materials scientist Della Marie Martin on June 8 that year.