The generalized k-connectivity κ_k(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by GˆH and G&Box;H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ₃(GˆH)≥ κ₃(G)|V(H)|. We also give upper bounds for κ₃(G&Box; H) and κ₃(GˆH). Moreover, all the bounds are sharp.