We have considered a pursuit game with one escapee and one pursuer. The managed conflictprocess is describedin the system of differentialdifference equations of a neutral type. Such equations contain an unknownfunction and its derivatives at different points of time and have not been applied in the theory of differential games yet. Effective in solving particular pursuit game tasks is a resolving functions method that is closely related to L. S. Pontryagin’s first direct method and commonly used in regular differential games and differential-difference games of a delayed type. We have devised a modified method of resolving functions for differentialdifference pursuit games of a neutral type. In the pursuit process, there exists a switch-over point that starts the catch time. This proves that the escapee’s errors do not affect the guaranteed time of the game end, which is calculated and set in advanceby the process parameters.The study has revealed adequate conditions for the process parameters that allow finishing the game within the fixed end time. The class of the known differential pursuit games can be expanded by the formulated pursuit task, whose process is described in the system of differentialdifference equations of a neutral type, and the devised scheme of the resolving functions method. This facilitates further consideration of such processes in the pursuit task with non-fixed time, objects of various inertia, and integral restrictions.