A measurement of angular correlations in Drell-Yan lepton pairs via the phi(eta)* observable is presented. This variable probes the same physics as the Z/gamma* boson transverse momentum with a better experimental resolution. The Z/gamma* -> e(+)e(-) and Z/gamma* -> mu(+)mu(-) decays produced in proton-proton collisions at a centre-of-mass energy of root s = 7 TeV are used. The data were collected with the ATLAS detector at the LHC and correspond to an integrated luminosity of 4.6 fb(-1). Normalised differential cross sections as a function of phi(eta)* are measured separately for electron and muon decay channels. These channels are then combined for improved accuracy. The cross section is also measured double differentially as a function of phi(eta)* for three independent bins of the Z boson rapidity. The results are compared to QCD calculations and to predictions from different Monte Carlo event generators. The data are reasonably well described, in all measured Z boson rapidity regions, by resummed QCD predictions combined with fixed-order perturbative QCD calculations or by some Monte Carlo event generators. The measurement precision is typically better by one order of magnitude than present theoretical uncertainties. (C) 2013 CERN. Published by Elsevier B.V. All rights reserved.