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