A search for supersymmetry in events with large missing transverse momentum, jets, and at least one hadronically decaying tau lepton has been performed using 3.2 fb(-1) of proton-proton collision data at root s = 13 TeV recorded by the ATLAS detector at the Large Hadron Collider in 2015. Two exclusive final states are considered, with either exactly one or at least two tau leptons. No excess over the Standard Model prediction is observed in the data. Results are interpreted in the context of gauge-mediated supersymmetry breaking and a simplified model of gluino pair production with tau-rich cascade decays, substantially improving on previous limits. In the GMSB model considered, supersymmetry-breaking scale (Alpha) values below 92 TeV are excluded at the 95% confidence level, corresponding to gluino masses below 2000 GeV. For large values of tan beta, values of Alpha up to 107 TeV and gluino masses up to 2300 GeV are excluded. In the simplified model, gluino masses are excluded up to 1570 GeV for neutralino masses around 100 GeV. Neutralino masses below 700 GeV are excluded for all gluino masses between 800 and 1500 GeV, while the strongest exclusion of 750 GeV is achieved for gluino masses around 1450 GeV.