The ordinal invariants, i.e., maximal order type, height, and width, are measures of a well quasi-ordering (wqo) based on the ordinal rank of the trees of its bad sequences, strictly decreasing sequences, and antichain sequences, respectively. Complex wqos are often built from simpler wqos through basic constructions such as disjoint sum, direct sum, cartesian product, and higher-order constructions like powerset or sequences. One main challenge is to compute the ordinal invariants of such wqos compositionally. This article focuses on the width of the cartesian product of wqos, for which no general formula is known. The particular case of the cartesian product of two ordinals has already been solved (Abraham, Order 4, 1987). We generalize this study and compute the width of the cartesian product of finitely many ordinals. To this end, we develop new tools for proving lower bounds on the width of wqos. Finally, we leverage our main result to compute the width of a generic family of elementary wqos that is closed under cartesian product.