He was born in Athens in 1907, the second son of Panaghiotis and Errietti Engonopoulos. During the summer of 1914, after a trip to Constantinople, his family was obliged to settle, due to the outbreak of World War I. In 1923, he was enrolled in a lycée in Paris, where he studied for a period of four years. After his return to Greece, he served as a private in the 1st Infantry Regiment and later on, he worked as a translator in a bank and as a secretary at the University of Athens. In 1930, Engonopoulos was employed as a designer in the Urban Planning Department of the Greek Ministry of Public Works.
In 1932, he joined the Athens School of Fine Arts, where he studied under the supervision of Konstantinos Parthenis. He also attended classes in the art studio of Photios Kontoglou. During that time, he met important artists, poet Andreas Embirikos and painters Yannis Tsarouchis, Giorgio de Chirico and Yannis Moralis being among them.
His first paintings, mostly temperas on paper, depicting old houses, were presented at an Art of Modern Greek Tradition exhibition, organised in January 1938. Soon after the exhibition, he published translations of poems by Tristan Tzara, which were published in February. A few months later, his first collection of poems (Do Not Distract the Driver) was published, followed by a second one (The Clavicembalos of Silence), the next year. Overall he is considered one of the finest surrealist poets of Greece.
His first individual exhibition was held in 1939. Three years later, he finished his most popular long poem Bolivar, a Greek Poem, inspired by the revolutionary leader Simón Bolívar and published in 1944. The poem was also released in the form of a song, in 1968, with music by composer Nikos Mamangakis.
He died of a heart attack in 1985.