Estimation of Lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control. Known results in this field are based on the Gronwall inequality yielding relatively conservative bounds for Lyapunov exponents. In this note, we obtained more accurate upper bounds for the general Lyapunov exponent for systems consisting of a known linear time-varying part and an unknown nonlinear component with a bounded Lipschitz constant at zero. Consequently, a sufficient condition for exponential stability of this system is formulated. The systems are indicated for which the obtained bound is precise, i.e., cannot be improved without additional information on the nonlinear term. In the presence of a persisting perturbation, an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system. Using the obtained results, a condition for exponential stability of a linear time-varying control system with a nonlinear feedback is derived. Numerical results are obtained for a second-order time-varying system and for the Lienard equation; in the latter case they are favorably compared with stability conditions previously obtained using the Lyapunov function method.