Theorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a time-delay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations.