This paper presents a new extremal-field approach for synthesizing nearly-optimal feedback controllers for optimal control and two-player pursuit-evasion games described by general non-linear differential equations. The proposed method utilizes a spatial statistical technique called universal kriging to construct the surrogate model of a feedback controller, which is capable of quickly predicting an optimal control estimate based on current state (and time) information. The kriging framework estimates an unknown function by observing its input-output behavior at a number of neighboring 'training' sites. In universal kriging, a feedback map is considered to be the linear combination of a deterministic regression polynomial model and Gaussian random function. The parameters of the kriging model are computed based on data from offline-computed open-loop extremals, generated by a direct or indirect method. Numerical examples involving both autonomous and nonautonomous systems are presented to evaluate the effectiveness of this methodology. For each solved problem, the performance of the feedback controller is ascertained by comparing the feedback solutions with open-loop ones. The numerical accuracy of the feedback solutions is very encouraging, and the low computational overhead of this approach makes it a suitable candidate for real-time applications.