Szulc was born in Warsaw, the son of Seweryn and Janina Baruch Szulc. He attended school in Switzerland. In 1940 he emigrated from Poland to join his family (who had left Poland in the mid-1930s) in Brazil. There he studied at the University of Brazil, but in 1945 he abandoned his studies to work as a reporter for the Associated Press in Rio de Janeiro.
In 1949 he moved to New York City, and in 1954 he became an American citizen.
Married for 52 years, he had a son and daughter.
Szulc died of hepatocellular carcinoma and lung cancer, aged 74.
From 1953 to 1972 Szulc was a foreign and Washington correspondent for The New York Times.
On 6 April 1961, nine days before the CIA-supported Bay of Pigs invasion in Cuba, Szulc wrote a New York Times article stating that an invasion of Cuba was "imminent." Prior to its publication, President Kennedy became aware of the article and personally telephoned the New York Times' publisher. The Times yielded to the President's demand that the story be reduced in prominence and detail.