The uninterrupted 7-day ASCA observations of the TeV blazar Mrk 421 in 1998 have clearly revealed that X-ray flares occur repeatedly. In this paper, we present the results of the time-resolved spectral analysis of the combined data taken by ASCA, RXTE, BeppoSAX, and EUVE. In this object--and in many other TeV blazars--the precise measurement of the shape of the X-ray spectrum, which reflects the high energy portion of the synchrotron component, is crucial in determining the high energy cutoff of the accelerated electrons in the jet. Thanks to the simultaneous broadband coverage, we measured the 0.1-25 keV spectrum resolved on time scales as short as several hours, providing a great opportunity to investigate the detailed spectral evolution at the flares.