This paper describes precision measurements of the transverse momentum \(p_\mathrm{T}^{\ell\ell}\) (\(\ell=e,\mu\)) and of the angular variable \(\phi^{*}_{\eta}\) distributions of Drell-Yan lepton pairs in a mass range of 66-116 GeV. The analysis uses data from 36.1 fb\(^{-1}\) of proton-proton collisions at a centre-of-mass energy of \(\sqrt{s}=13\) TeV collected by the ATLAS experiment at the LHC in 2015 and 2016. Measurements in electron-pair and muon-pair final states are performed in the same fiducial volumes, corrected for detector effects, and combined. Compared to previous measurements in proton-proton collisions at \(\sqrt{s}=\)7 and 8 TeV, this new measurement probes perturbative QCD at a higher centre-of-mass energy with a different composition of initial states. It reaches a precision of 0.2% for the normalized spectra at low values of \(p_\mathrm{T}^{\ell\ell}\). The data are compared with different QCD predictions, where it is found that predictions based on resummation approaches can describe the full spectrum within uncertainties.