Yevgeny Matveyev was born in the village of Novoukrainka in the Mykolaiv Governorate of the Ukrainian Soviet Socialist Republic (now Kherson Oblast, Ukraine) to father Semyon Kalinovich Matveyev, a Red Army serviceman was stationed in the region at the end of the Russian Civil War, and mother Nadezhda Fyodorovna Kovalenko, a local peasant woman. Semyon Matveyev left his wife shortly after Yevgeny was born.
Yevgeny Matveyev attended school in the nearby town of Tsyurupinsk, where he saw his first play and left school after the ninth grade to pursue a career in acting.