The dark photon (A′) production through the mixing with the bremsstrahlung photon from the electron scattering off nuclei can be accompanied by the dominant invisible A′ decay into dark-sector particles. In this work we discuss the missing energy signature of this process in the experiment NA64 aiming at the search for A′→invisible decays with a high-energy electron beam at the CERN SPS (The Super Proton Synchrotron). We show the distinctive distributions of variables that can be used to distinguish the A′→invisible signal from background. The results of the detailed simulation of the detector response for the events with and without A′ emission are presented. The efficiency of the signal event selection is estimated. It is used to evaluate the sensitivity of the experiment and show that it allows us to probe the still unexplored area of the mixing strength 10−6≲ε≲10−2 and masses up to MA′≲1 GeV. The results obtained are compared with the results from other calculations. In the case of the signal observation, a possibility of extraction of the parameters MA′ and ε by using the shape of the missing energy spectrum is discussed.