The notions of quasi-finitely axiomatizable (QFA) groups and quasi-axiomatizable (QA) groups were introduced by A. Nies in 2003. We say that a finitely generated group G is QFA if there exists a first-order sentence ϕ that is satisfied by G such that each finitely generated group that satisfies ϕ is isomorphic to G. The aim of this paper is to further investigate properties of direct products with respect to these notions, thereby extending F. Oger's result for nilpotent-by-finite groups. More precisely, we study the relations between (1) G and H are QFA, and (2) G×H is QFA. We prove that (1) implies (2) under the additional condition that there exist bounds on the commutator width in the central extensions of the center Z(H) by G and Z(G) by H. We also prove that the converse is true if the groups are ℤ-replaceable, in particular if they are residually soluble. Thus (1) and (2) are equivalent, for instance, in the classes of finitely generated abelian-by-polycyclic or abelian-by-metanilpotent groups.