Prabhu, who was from Mangalore and Kozhikode got a B.A. in mathematics from University of Madras (1946), a M.A. in statistics from University of Bombay (1950). Prabhu was the founding head of the department of statistics, Karnatak University, Dharwar in 1951.^[2] He obtained a M.Sc. in mathematics from University of Manchester on a thesis entitled Solution to Some Dam Problems (1957).

Prabhu lectured at Gauhati University (1950–1952), Karnatak University (1952–1961), University of Western Australia (1961–1964), before becoming associate professor at University of Michigan (1964–1965) and Cornell University (1965–1994) where he became professor (1967) and emeritus (1994). Prabhu also had longer research stays at Indian Statistical Institute in Calcutta (1961), University of Wisconsin–Madison (1970, 1973), Technion in Haifa (1973), University of Melbourne (1978), University of Maryland, College Park (1979), and Uppsala University (1984) ^{[citation needed]}.

Prabhu was the founding editor of the Queueing Systems (journal) (1986–1994) and has edited several other journals, as well as published books on Foundations of Queueing Theory (Springer Verlag, 1997) and Stochastic storage processes (Springer, 1998).

The South Asia Program at Cornell created the Rabindranath Tagore Endowment in Modern Indian Literature, made possible through a gift by Professor Emeritus Narahari Umanath Prabhu and his wife ^{[citation needed]}.

Prabhu died on 14 October 2022, at the age of 98.^[3]

• INFORMS (Applied Probability) Award of Honor (1997)
• Honorary fellow of the Indian statistical association (1998)

• N. U. Prabhu at the Mathematics Genealogy Project
• On Google scholar