Robust control over the dynamics of a quantum system is indispensable for realizing useful quantum information processors. Here, we utilize the classical method of averaging to realize an approach for robustness analysis and synthesis of controls for quantum logic gates. By separating the system Hamiltonian into a nominal (uncertainty-free) and interaction (error) Hamiltonian, the robustness measure is defined as the size of the average of the interaction Hamiltonian. The result is a multi-criterion optimization competing uncertainty-free fidelity with the robustness measure. Exploiting the topology of the control landscape at high fidelity (as determined by the null space of the nominal fidelity Hessian), we arrive at a two-stage optimization algorithm: first, improve the nominal fidelity until sufficiently high, and then reduce the robustness measure while maintaining high nominal fidelity. An optimal solution is obtained by approximating the nominal fidelity and robustness measure as quadratics in the control increments. The approximation enables the control increments at each iteration to be solved by a convex optimization. For each iteration, the goal is to keep the nominal fidelity high while reducing the robustness measure. This approach reduces the experimental quantum-system-characterization effort, enabling robust and high-fidelity quantum logic gate design.