The authors obtain sufficient conditions for the stability of the zero solution of the essentially nonlinear system \[ \dot{x}(t)=f(x(t),t) \tag{1} \] when the asymptotic stability of the zero solution of the averaged system \[ \dot{x}(t)=\bar f(x(t)) \tag{2} \] yields the local uniform asymptotic stability of the complete system (1). Here, \(f\) is T-periodic in \(t\), \(f(x,t)\equiv f(x,t+T)\), \(\bar f(x)=\frac{1}{T}\int_0^Tf(x,t)\, dt\). The result is illustrated with several examples. Reviewer's remark: The main theorem follows from a more general result presented in the paper by the reviewer [Differ. Equations 14, 1061--1063 (1978; Zbl 0411.34066)].