The azimuthal anisotropy of charged particles in Pb-Pb collisions at sqrt[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 (v2) 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 v2 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 pp(T)=40 GeV/c over the full centrality range measured.