A search for the associated production of the Higgs boson with a top quark pair (ttH) is reported. The search is performed in multilepton final states using a data set corresponding to an integrated luminosity of 36.1 fb-1 of proton-proton collision data recorded by the ATLAS experiment at a center-of-mass energy s=13 TeV at the Large Hadron Collider. Higgs boson decays to WW∗, ττ, and ZZ∗ are targeted. Seven final states, categorized by the number and flavor of charged-lepton candidates, are examined for the presence of the Standard Model Higgs boson with a mass of 125 GeV and a pair of top quarks. An excess of events over the expected background from Standard Model processes is found with an observed significance of 4.1 standard deviations, compared to an expectation of 2.8 standard deviations. The best fit for the ttH production cross section is σ(ttH)=790-210+230 fb, in agreement with the Standard Model prediction of 507-50+35 fb. The combination of this result with other ttH searches from the ATLAS experiment using the Higgs boson decay modes to bb, γγ and ZZ∗→4, has an observed significance of 4.2 standard deviations, compared to an expectation of 3.8 standard deviations. This provides evidence for the ttH production mode.