He was born in Biysk to a family of a Moscow engineer on a business trip. The family returned to Moscow fairly soon.
"Genrikh Sapgir is the most prominent figure of the writers that came to be associated with the now well-known Lianozovo group (ru:Лианозовская школа), which also included Vsevolod Nekrasov and Igor Kholin. These Moscow poets sought out new models and positions and exploited the possibilities of inserting common speech directly in their texts. Each of them had a Dostoyevskian eye for everyday Russian life, which made their work immediately accessible."
Since 1959 he published his poetry for children. His other poems appeared only in émigré magazines, such as Continent and Strelets (The Archer).
During the perestroika period
Since 1989 his poetry, short stories, plays and novels have been widely published in Russia. Three volumes of his Collected Poems appeared at the end of the 1990s. He represented Russia at numerous international festivals of poetry, his work has been published in translation throughout the world. There are English translations by Jim Kates, Anatoly Kudryavitsky  and Artyom Kotenko & Anthony Weir  Andrew Bromfield published his translations of Sapgir's 'Very Short Stories'. Sapgir was the recipient of various awards including the Pushkin Prize for poetry.
In 1999 he died of a heart attack in a Moscow trolley-bus on his way to the launch of the anthology of contemporary Russian poetry entitled "Poetry of Silence".
Books in English Translation
- Psalms. Translated by Jim Kates Cold Hub Press, 2012.
- Kudryavitsky, A. A Night in the Nabokov Hotel: 20 Contemporary Poets from Russia
- Shrayer, M., Shrayer-Petrov, D. Генрих Сапгир: Классик авангарда. [Genrikh Sapgir: An avant-garde classic]. Dmitri Bulanin Publishing, St Petersburg, 2004
- Smith, A. 'Genrikh Sapgir: Klassik avangarda.' The Slavonic and East European Review, Volume 83, Number 4, 1 October 2005, pp. 746–747(2)
- Kudryavitsky, A. A Night in the Nabokov Hotel: 20 Contemporary Poets from Russia, Introduction, pp. 1–2.
- Official website (in Russian)
- Very short stories in English translated by Andrew Bromfield
- Eight poems in English translated by Artyom Kotenko and Anthony Weir
- Five poems in English translated by Anatoly Kudryavitsky
- "Psalm 136" translated by Roman Turovsky
- Ivan Karamazov's interview with Anatoly Kudryavitsky about Sapgir (in Russian)