A measurement of production cross sections of the Higgs boson in proton-proton collisions is presented in the H -> 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 root s = 13 TeV. All combinations of leptonic (tau -> l v (v) over bar with l = e, mu) and hadronic (tau -> hadrons v) tau decays arc considered. The H -> 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 -> tau tau decays. Using the data taken at root s = 13 TeV, the total cross section in the H -> tau tau decay channel is measured to be 3.77(-0.59)(+0.60)(stat)(-0.74)(+0.87) (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 -> tau tau decay channel are determined separately for vector-boson-fusion production and gluon-gluon-fusion production to be sigma(VBF)(H -> tau tau) = 0.28 +/- 0.09 (stat)(-0.09)(+0.11) (syst) pb and sigma(ggF)(H -> tau tau) = 3.1 +/- 1.0 (stat)(-1.3)(+1.6) (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.