Measurements of the Higgs production cross section in the $H \rightarrow \tau \tau$ decay channel are presented. The analysis has been performed using $36.1$ fb$^{-1}$ of data collected by the ATLAS experiment at $\sqrt{s}=13$ TeV proton-proton collisions at the Large Hadron Collider. The observed (expected) significance of the $H \rightarrow \tau \tau$ signal excess over the expected background amounts to 4.4 (4.1) standard deviations. This result, combined with the data taken at $\sqrt{s}=7$ and 8 TeV, leads to an observed (expected) significance of 6.4 (5.4) and constitutes the first ATLAS observation of $H \rightarrow \tau \tau$. The measured total cross section of $H \rightarrow \tau \tau$, using the data collected at $\sqrt{s}=13$ TeV, is $3.77^{+0.60}_{-0.59}~(\text{stat.})~^{+0.87}_{-0.74}~(\text{syst.})$ pb, assuming the relative contributions of the Higgs production processes as predicted by the Standard Model. In addition, total cross sections for the vector boson fusion and gluon-gluon fusion processes have been measured separately and similar results are reported based on the simplified template cross section framework. All the measurements are in agreement with the Standard Model predictions.