Measurements of charge-dependent azimuthal correlations with the ALICE detector at the LHC are reported for Pb-Pb collisions at sqrt[s(NN)] = 2.76 TeV. Two- and three-particle charge-dependent azimuthal correlations in the pseudorapidity range |η| < 0.8 are presented as a function of the collision centrality, particle separation in pseudorapidity, and transverse momentum. A clear signal compatible with a charge-dependent separation relative to the reaction plane is observed, which shows little or no collision energy dependence when compared to measurements at RHIC energies. This provides a new insight for understanding the nature of the charge-dependent azimuthal correlations observed at RHIC and LHC energies.