A measurement of the Standard Model Higgs boson produced in association with a $W$ or $Z$ boson and decaying into a pair of $\tau$-leptons is presented. This search is based on proton-proton collision data collected at $\sqrt{s}=13$ TeV by the ATLAS experiment at the LHC corresponding to an integrated luminosity of 140 fb$^{-1}$. For the Higgs boson candidate, only final states with at least one $\tau$-lepton decaying hadronically ($\tau\rightarrow \mathrm{hadrons} + \nu_\tau$) are considered. For the vector bosons, only leptonic decay channels are considered: $Z \rightarrow \ell\ell$ and $W\rightarrow \ell\nu_\ell$, with $\ell=e,\mu$. An excess of events over the expected background is found with an observed (expected) significance of 4.2 (3.6) standard deviations, providing evidence of the Higgs boson produced in association with a vector boson and decaying into a pair of $\tau$-leptons. The ratio of the measured cross-section to the Standard Model prediction is $\mu_{\text{VH}}^{\tau\tau} = 1.28\ ^{+0.30}_{-0.29}\ (\mathrm{stat.})\ ^{+0.25}_{-0.21}\ (\mathrm{syst.})$. This result represents the most accurate measurement of the \vh process achieved to date.