A measurement of production cross sections of the Higgs boson in proton-proton collisions is presented in the $H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}$ decay channel. The analysis is performed using $36.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of data recorded by the ATLAS experiment at the Large Hadron Collider at a center-of-mass energy of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$. All combinations of leptonic ($\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\ell}v\overline{v}$ with $\ensuremath{\ell}=e,\ensuremath{\mu}$) and hadronic ($\ensuremath{\tau}\ensuremath{\rightarrow}\text{hadrons}\text{ }v$) $\ensuremath{\tau}$ decays are considered. The $H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}$ signal over the expected background from other Standard Model processes is established with an observed (expected) significance of 4.4 (4.1) standard deviations. Combined with results obtained using data taken at 7 and 8 TeV center-of-mass energies, the observed (expected) significance amounts to 6.4 (5.4) standard deviations and constitutes an observation of $H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}$ decays. Using the data taken at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$, the total cross section in the $H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}$ decay channel is measured to be $3.7{7}_{\ensuremath{-}0.59}^{+0.60}\text{ }(\mathrm{stat}){\text{ }}_{\ensuremath{-}0.74}^{+0.87}\text{ }(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, for a Higgs boson of mass 125 GeV assuming the relative contributions of its production modes as predicted by the Standard Model. Total cross sections in the $H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}$ decay channel are determined separately for vector-boson-fusion production and gluon-gluon-fusion production to be ${\ensuremath{\sigma}}_{H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}}^{\mathrm{VBF}}=0.28\ifmmode\pm\else\textpm\fi{}0.09\text{ }(\mathrm{stat}){\text{ }}_{\ensuremath{-}0.09}^{+0.11}\text{ }(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$ and ${\ensuremath{\sigma}}_{H\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau}}^{gg\mathrm{F}}=3.1\ifmmode\pm\else\textpm\fi{}1.0\text{ }(\mathrm{stat}){\text{ }}_{\ensuremath{-}1.3}^{+1.6}\text{ }(\mathrm{syst})\text{ }\text{ }\mathrm{pb}$, respectively. Similarly, results of a fit are reported in the framework of simplified template cross sections. All measurements are in agreement with Standard Model expectations.