A measurement of production cross sections of the Higgs boson in proton-proton collisions is presented in the $H\to \tau \tau$ decay channel. The analysis is performed using $36.1\mathrm{fb}-1$ of data recorded by the ATLAS experiment at the Large Hadron Collider at a center-of-mass energy of $s=13$ TeV. All combinations of leptonic ($\tau \to l\nu \nu ¯$ with $l=e$, $\mu$) and hadronic ($\tau \to \mathrm{hadrons}\nu$) tau decays are considered. The $H\to \tau \tau$ 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 $H\to \tau \tau$ decays. Using the data taken at $s=13$ TeV, the total cross section in the $H\to \tau \tau$ decay channel is measured to be $3.77-0.59+0.60(\mathrm{stat}.)-0.74+0.87(\mathrm{syst}.)\mathrm{pb}$, 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 $H\to \tau \tau$ decay channel are determined separately for vector-boson-fusion production and gluon-gluon-fusion production to be $0.28\pm 0.09(\mathrm{stat}.)-0.09+0.11(\mathrm{syst}.)\mathrm{pb}$ and $3.1\pm 1.0(\mathrm{stat}.)-1.3+1.6(\mathrm{syst}.)\mathrm{pb}$, 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.