Measurements of the Higgs production cross section in the $H \rightarrow \tau \tau$ decay channel are presented. The results are obtained using $36.1$ fb$^{-1}$ of data collected by the ATLAS experiment at $\sqrt{s}=13$ TeV. 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, measurements of the vector boson fusion and gluon-gluon fusion have been performed 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.