Grosswald was born on December 15, 1912 in Bucharest, Romania. He received a Master's degree from the University of Bucharest in 1933.

Grosswald was Jewish, and fled from the Nazis to Paris in the late 1930s, then to Orléans, then through Spain to Cuba where he spent the rest of World War II. He moved to Puerto Rico in 1946 and then to the United States in 1948. He received his PhD under Hans Rademacher from the University of Pennsylvania in 1950.

Grosswald died April 11, 1989 in Narberth, Pennsylvania.

Grosswald's first three scientific papers, written while he was in Cuba, were published under the pseudonym E.G. Garnea.

After receiving his PhD in 1950, Grosswald taught at the University of Pennsylvania from 1952 to 1968 and then moved to Temple University and stayed until his retirement in 1980. He also held positions at the University of Saskatchewan (1950), Institute for Advanced Study (1951), the Technion (1980 – 1981), Swarthmore College (1982), and the University of Pennsylvania (1984).

Grosswald completed some works of his teacher Hans Rademacher, who died in 1969. Rademacher had prepared notes for an Earle Raymond Hedrick Lecture in Boulder, Colorado, in 1963 on Dedekind sums, but fell ill, and Grosswald gave the lecture for him. After Rademacher's death, Grosswald edited and completed the notes and published them in the Carus Mathematical Monographs series as Dedekind Sums. He also edited for publication Rademacher's posthumous textbook Topics in Analytic Number Theory.

Grosswald was elected to the Board of Governors of the Mathematical Association of America for 1965 – 1968. Temple University's Mathematics Department annually sponsors the Emil Grosswald Memorial Lectures.