The azimuthal anisotropy of charged particles in Pb-Pb collisions at root S-NN = 2.76 TeV is measured with the CMS detector at the LHC over an extended transverse momentum (p(T)) range up to approximately 60 GeV/c. The data cover both the low-p(T) region associated with hydrodynamic flow phenomena and the high-p(T) region where the anisotropies may reflect the path-length dependence of parton energy loss in the created medium. The anisotropy parameter (nu(2)) of the particles is extracted by correlating charged tracks with respect to the event-plane reconstructed by using the energy deposited in forward-angle calorimeters. For the six bins of collision centrality studied, spanning the range of 0-60\% most-central events, the observed nu(2) values are found to first increase with p(T), reaching a maximum around p(T) = 3 GeV/c, and then to gradually decrease to almost zero, with the decline persisting up to at least p(T) = 40 GeV/c over the full centrality range measured.