The energy spread of the electron beam is a critical parameter in x-ray free-electron lasers (XFELs) and needs to be optimized for best performance. The uncorrelated energy spread of the electrons can be a few keV or less in XFEL injectors, thus very challenging to measure. The standard method to characterize the electron beam energy spread, consisting in streaking the beam with a transverse deflector and measuring the time-resolved beam size of the electrons in a dispersive location for a single electron beam energy, has a typical resolution of several keV. To overcome this limitation we introduce a novel method to measure the beam size at a dispersive location for different beam energies so that it is possible to disentangle the beam size contributions related to the energy spread, the intrinsic beam size and the monitor resolution. As a consequence, the energy spread can be characterized with a much higher precision and resolution than in the standard approach. We also suggest to perform measurements for different deflection amplitudes so that the energy spread induced by the transverse deflector can be subtracted properly. The scheme does not require any additional hardware and thus can be readily applied in any standard XFEL facility. Numerical simulations and experimental results at SwissFEL confirm the validity of our method. Our calculations show that the approach can be used to significantly overcome the resolution of the standard approach and measure energy spreads well below 1 keV. As an example we present energy spreads of few keV measured at the SwissFEL injector.