Veneta is a city in Lane County, Oregon, United States. As of the 2010 census, the city population was 4,561. Veneta was established in 19…Veneta is a city in Lane County, Oregon, United States. As of the 2010 census, the city population was 4,561. Veneta was established in 1912 by Edmund Eugene Hunter, who named the settlement after his five-year-old daughter. Veneta post office was established in 1914. Veneta Hunter Vincent, the city's namesake, died in 2000 at age 91. She had attended the city's 70th anniversary party in 1982.