This paper describes precision measurements of the transverse momentum p(T)(ll) (l = 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) ovf proton-proton collisions at a centre-of-mass energy of root 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 root s = 7 and 8 TeV, these new measurements probe perturbative QCD at a higher centre-of-mass energy with a different composition of initial states. They reach a precision of 0.2% for the normalized spectra at low values of p(T)(ll). 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.