The method for synthesis of closed loop quasi-time-optimal high-order (n-order) control systems is developed for a wide class of objects with polynomial and rational nonlinearities. This method is based on the transition from the problem of time-optimal control to the corresponding problem of analytical design of an optimal controller for the considered object using the specifically given performance index. Well-known A. A. Krasovsky's power series method is modified to solve the latter problem, that allows to obtain the sequence of approximations to the optimal control with increasing accuracy. The proposed synthesis method has the following principal features: 1) the representation of the unit function in the time-optimal criterion by a positively defined polynomial or rational function, that ensures the smoothness of a Bellman function and applicability of the dynamic programming method to the solution of the time-optimal control problem with a significantly lower amount of computations (n times reduction for a n-th order object); 2) the control law is defined in the form of a rational expression of object coordinates, that significantly increases accuracy of optimal control approximation in comparison with A. A. Krasovsky's method under the same number of parameters; 3) the synthesis method has an iterative character and allows a relatively simple implementation; 4) the method is applicable for solution of both problems of time-optimal and energy-saving control design; 5) the obtained control is continuous, not relay one, which eliminates sliding and oscillatory modes that are undesirable for considered objects.