Fast radio bursts (FRBs) are short pulses observed in radio band from cosmological distances, some of which emit repeating bursts. The physical origins of these mysterious events have been subject to wide speculations and heated debates. One class of models invoke soft gamma-ray repeaters (SGRs), or magnetars, as the sources of FRBs. Magnetars are rotating neutron stars with extremely strong magnetic field and can sporadically emit bursts from X-ray (keV) to soft gamma-ray (sub-MeV) with duration from 10􀀀2 s to 102 s. However, even though some bright radio bursts have been observed from some magnetars, no FRB-like events had been detected to be associated with any magnetar burst, including one giant flare, and no radio burst has been associated with any X-ray event from any magnetar. Therefore, there is still no observational evidence for magnetar-FRB association up to today. Recently, a pair of FRB-like bursts (FRB~200428 hereafter) separated by 30 milliseconds (ms) were detected from the general direction of the Galactic magnetar SGR~J1935+2154. Here we report the detection of a non-thermal X-ray burst in the 1--250\,keV energy band with the Insight-HXMT satellite, which we identify as emitted from SGR~J1935+2154. The burst showed two hard peaks with a separation of ms, consistent with the separation between the two bursts in FRB~200428. The delay time between the double radio and X-ray peaks is 8:57s, fully consistent with the dispersion delay of FRB~200428. We thus identify the non-thermal X-ray burst is associated with FRB~200428 whose high energy counterpart is the two hard peaks in X-ray. Our results suggest that the non-thermal X-ray burst and FRB~200428 share the same physical origin in an explosive event from SGR~J1935+2154.