This paper outlines a method for solving the game problem of convergence of the generalized quasi-linear non-stationary system trajectory with the cylindrical terminal set whose body part depends on time. The method of resolving functions stands for the theoretical basis of investigation. We examine the situation when Pontryagin’s condition, reflecting an advantage in control resources of one of the opposing sides, does not hold. Because of this, with the help of a certain scalar function, we construct the modified condition. This function is called the function of time dilation. Fulfillment of this condition provides an opportunity for the pursuer, based on the Filippov-Castaing theorems on measurable choice, to build control which guarantees successive solution of the problem at hand. In so doing, the technique of set-valued mapping and their selections is applied. The process of convergence consists of two parts – active and passive. Control of the first player is constructed in view of his opponent’s control in the past, namely with a certain time delay depending on the function of time dilation. We obtain sufficient conditions for convergence in a finite time in the class of quasi-strategies, and, under additional conditions, in the class of stroboscopic strategies with delay.