Panjabi's grandparents were refugees from Sindh Province following the British Partition of India and Pakistan in 1947, resettling in Mumbai and Indore in India. A generation later, Panjabi's parents migrated to West Africa, where Panjabi was born and raised in Monrovia, Liberia. After civil war broke out in Liberia in 1989, Panjabi, at age nine, and his family fled on a rescue cargo plane to Sierra Leone and eventually sought asylum in the United States, resettling initially with a host family in High Point, North Carolina.