We analyze the second-class current decays $\tau^-\to\pi^-\eta(')\nu_\tau$ in the framework of Chiral Perturbation Theory with Resonances. Taking into account $\pi^0-\eta-\eta'$ mixing, the $\pi-\eta(')$ vector form factor is extracted, in a model-independent way, using existing data on the $\pi-\pi_0$ one. For the participant scalar form factor, we have considered different parameterizations ordered according to their increasing fulfillment of analyticity and unitarity constraints. We start with a Breit-Wigner parameterization dominated by the $a_0(980)$ scalar resonance and after we include its excited state, the $a_0(1450)$. We follow by an elastic dispersion relation representation through the Omnes integral. Then, we illustrate a method to derive a closed-form expression for the $\pi-\eta$, $\pi-\eta'$ (and $K−K^0$) scalar form factors in a coupled-channels treatment. Finally, predictions for the branching ratios and spectra are discussed emphasizing the error analysis. An interesting result of this study is that both $\tau^-\to\pi^-\eta(')\nu_\tau$ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner $\eta(')_{l3}$ decays, which are extremely suppressed in the Standard Model.