Katznelson was born in Bobruisk in the Russian Empire (now in Belarus) to Nissan HaCohen and Zelda Helena (née Rozovsky).

He held additional public positions, such as being on the Technion's executive committee and the Palestine Broadcasting Service's advisory council during the Mandate era.

David Ben-Gurion noted in his diary after meeting with the establishment team: "An emergency medical committee has been formed, and regional doctors have been appointed.

"[10]Katznelson was a member of the Provisional State Council due to his role in the leadership of the National Committee,[11] and was among the signatories of the Israeli Declaration of Independence.

He was the father of Shmuel (a doctor, professor and a pioneer in paediatric surgery in Israel) and Ze'evah (Zika), the wife of Matti Peled.

His memory is commemorated with a street named "Avraham Nissan" in the Kiryat Yovel neighborhood in Jerusalem.