Agajanian was born in San Pedro, Los Angeles, California, just six months after his father had immigrated to the United States out of Armenia. He grew up a hard-working young man in the family's refuse collection and hog ranching businesses, which "Aggie" would later oversee.

From 1948 through 1971, his cars won three pole positions – the first with Walt Faulkner in 1950 – for the Indianapolis 500, set four track records and won the race twice. Troy Ruttman (1952) and Parnelli Jones (1963) both won the 500 in Agajanian machines.