A search for the associated production of the Higgs boson with a top quark pair ($t\overline{t}H$) is reported. The search is performed in multilepton final states using a data set corresponding to an integrated luminosity of $36.1{\mathrm{fb}}^{-1}$ of proton-proton collision data recorded by the ATLAS experiment at a center-of-mass energy $\sqrt{s}=13\mathrm{TeV}$ at the Large Hadron Collider. Higgs boson decays to $W{W}^{*}$, $\tau \tau$, and $Z{Z}^{*}$ 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 $t\overline{t}H$ production cross section is $\sigma \left(t\overline{t}H\right)=790_{-210}^{+230}\mathrm{fb}$, in agreement with the Standard Model prediction of $507_{-50}^{+35}\mathrm{fb}$. The combination of this result with other $t\overline{t}H$ searches from the ATLAS experiment using the Higgs boson decay modes to $b\overline{b}$, $\gamma \gamma$ and $Z{Z}^{*}\to 4\ell$, has an observed significance of 4.2 standard deviations, compared to an expectation of 3.8 standard deviations. This provides evidence for the $t\overline{t}H$ production mode.