A class of systems of homogeneous differential-difference equations with linearly increasing delay is considered. In this class the matrix with the component without delay has a diagonal form. This kind of matrix allows us to investigate stability using the second Lyapunov method. Razumikhin’s approach is taken as a generalization of this method to differential-difference equations. Using this approach, we obtain conditions on the coefficients of the system of equations guaranteeing the asymptotic stability of the zero solution. In addition, two auxiliary lemmas are presented on changing the norm of a vector during term term exponentiation.