In this paper, it is shown that with large probability, the spectral radius of a large non-Hermitian random matrix with a general variance profile does not exceed the square root of the spectral radius of the variance profile matrix. A minimal moment assumption is considered and sparse variance profiles are covered. Following an approach developed recently by Bordenave, Chafa{ï} and Garc{í}a-Zelada, the key theorem states the asymptotic equivalence between the reverse characteristic polynomial of the random matrix at hand and a random analytic function which depends on the variance profile matrix. The result is applied to the case of a non-Hermitian random matrix with a variance profile given by a piecewise constant or a continuous non-negative function, the inhomogeneous (centered) directed Erdős-R{é}nyi model, and more.