We study the ballistic-to-diffusive transition induced by the weak breaking of integrability in a boundary-driven XXZ spin chain. Studying the evolution of the spin current density Js as a function of the system size L, we show that, accounting for boundary effects, the transition has a nontrivial universal behavior close to the XX limit. It is controlled by the scattering length L∗∝V−2, where V is the strength of the integrability-breaking term. In the XXZ model, the interplay of interactions controls the emergence of a transient “quasiballistic” regime at length scales much shorter than L∗. This parametrically large regime is characterized by a strong renormalization of the current which forbids a universal scaling, unlike the XX model. Our results are based on matrix product operator numerical simulations and agree with perturbative analytical calculations.