JOHANNESBURG—The U.S. Treasury froze assets owned by three brothers at the center of a sprawling government-corruption scandal in South Africa that contributed to the ouster of former President Jacob Zuma and ensnared major international companies.

Ajay, Atul and Rajesh Gupta, who arrived in South Africa from India in the early 1990s and quickly forged connections within the ruling African National Congress, used their ties to engage in corruption and bribery to get government contracts and misappropriate state assets, the...