In this paper, we investigate nonlinear Hamiltonian elliptic system {(-Δ)u=b→(x)⋅∇u+(V(x)+τ)u=K(x)g(v)inℝℕ,(-Δ)v=b→(x)⋅∇v+(V(x)+τ)v=K(x)f(u)inℝℕ,u(x)→0andv(x)→0as|x|→∞,where N ≥ 3, τ > 0 is a positive parameter and V, K are nonnegative continuous functions, f and g are both superlinear at 0 with a quasicritical growth at infinity. By establishing a variational setting, the existence of ground state solutions is obtained.