Saito (J Fac Sci Univ Tokyo Sect IA Math 27:265-291, 1980) gave a nice and efficient criterion to determine whether the module of logarithmic derivation associated with a reduced divisor in a complex variety is free or not. The aim of this note is to propose a new proof of this criterion, in the affine space, in the projective space, and for multiderivations, based on straightforward observations concerning free and reflexive modules. This point of view also allows us to prove a generalized version of the Saito criterion that applies to derivation modules associated with several polynomials introduced in Faenzi et al. (Logarithmic sheaves of complete intersections, 2021. arXiv:mathAG/2106.14453).