Magnetars, often under the name soft gamma-ray repeaters (SGRs) or anomalous X-ray pulsars, are highly magnetized neutron stars that exhibit diverse X-ray activities. Recently, a unique non-thermal X-ray burst with cut-off energy up to 84 keV is detected and thought to be associated with the fast radio burst (FRB) 200428 in the same single explosive event from SGR J1935+2154, as their spectra show similar feature of narrow double peaks that are emitted almost simultaneously. However, the physical origin of this FRB 200428-associated X-ray burst is still unknown yet. Here, with the first cross-scale numerical simulation in which modeling of particle acceleration by magnetic reconnections is self-consistently coupled with that of photon emission by multiple Compton scatterings, we identify that magnetic reconnection at the quantum-electrodynamic field strength inside the magnetar magnetosphere is the much likely driving source of such FRB-associated non-thermal X-ray burst. Both its temporal and spectral features are well reproduced in our simulations by assuming the plasma magnetization parameter σ~102-103 in consistency with the astronomical observations. The results could greatly promote our understandings of various X-ray burst events from magnetars.