Narasimhan was born on 7 June 1932 into a rural family in Tandarai in present day Tamil Nadu, as the eldest among five children.

[3] Narasimhan started his career in 1960 when he joined the faculty of the Tata Institute of Fundamental Research (TIFR); he later went on to become an honorary fellow.

[3] During this time, he visited France under the invitation of Laurent Schwartz and was exposed to the works of other French mathematicians including Jean-Pierre Serre, Claude Chevalley, Élie Cartan, and Jean Leray.

[3] During his time in France he also collaborated with Japanese mathematician Takeshi Kotake working on the analyticity theorems for determining specific types of elliptic operators that satisfied Cauchy–Schwarz inequalities.

[3][6] He collaborated with Indian mathematician C. S. Seshadri for the ground-breaking Narasimhan–Seshadri theorem which has been at the core of algebraic geometry and number theory for over half a century.

He also collaborated with mathematician R. R. Simha on proving the existence of moduli of general type complex structures on a real analytic manifold.

[3] In 1992, Narasimhan retired from TIFR, and became the head of the research group in Mathematics at the International Centre for Theoretical Physics in Trieste.

[12] He was also the recipient of the King Faisal International Prize for Science in 2006, an award that he won jointly with mathematician Simon Donaldson, Imperial College.

The couple had a daughter, Shobhana Narasimhan, a scientist and professor at Jawaharlal Nehru Centre for Advanced Scientific Research, and a son.