External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.