In this paper, a modification of the Krasovskii theorem for a nonlinear class of systems is presented. The idea is to replace the positive definiteness condition of the functional with this condition on the special Razumikhin-type set of functions only while retaining the other classical conditions. The result is motivated by the fact that this idea leads to the necessary and sufficient stability condition for linear time-invariant systems. Moreover, this condition is constructive and allows us not only to directly analyze the stability but also to find the robustness bounds on the matrix parameters and on the delays and to construct the exponential estimates for solutions. An overview of these results for linear systems is also presented.