We report the measurements of correlations between event-by-event fluctuations of amplitudes of anisotropic flow harmonics in nucleus-nucleus collisions, obtained for the first time using a new analysis method based on multiparticle cumulants in mixed harmonics. This novel method is robust against systematic biases originating from nonflow effects and by construction any dependence on symmetry planes is eliminated. We demonstrate that correlations of flow harmonics exhibit a better sensitivity to medium properties than the individual flow harmonics. The new measurements are performed in Pb-Pb collisions at the center-of-mass energy per nucleon pair of sqrt[s_{NN}]=2.76 TeV by the ALICE experiment at the Large Hadron Collider. The centrality dependence of correlation between event-by-event fluctuations of the elliptic v_{2} and quadrangular v_{4} flow harmonics, as well as of anticorrelation between v_{2} and triangular v_{3} flow harmonics are presented. The results cover two different regimes of the initial state configurations: geometry dominated (in midcentral collisions) and fluctuation dominated (in the most central collisions). Comparisons are made to predictions from Monte Carlo Glauber, viscous hydrodynamics, ampt, and hijing models. Together with the existing measurements of the individual flow harmonics the presented results provide further constraints on the initial conditions and the transport properties of the system produced in heavy-ion collisions.