New BeppoSAX observations of the nearby archetypical starburst galaxies (SBGs) NGC253 and M82 are presented. The main observational result is the unambiguous evidence that the hard (2-10 keV) component is (mostly) produced in both galaxies by thermal emission from a metal-poor (~ 0.1-0.3 solar), hot (kT ~ 6- 9 keV) and extended (see companion paper: Cappi et al. 1998) plasma. Possible origins of this newly discovered component are briefly discussed. A remarkable similarity with the (Milky Way) Galactic Ridges X-ray emission suggests, nevertheless, a common physical mechanism.