We present the most complete study to date of the X-ray emission from star formation in high-redshift (median z = 0.7; z \textless 1.5), IR-luminous (L-IR = 10(10)-10(13) L circle dot) galaxies detected by Herschel's PACS and SPIRE instruments. For our purpose, we take advantage of the deepest X-ray data to date, the Chandra Deep Fields (North and South). Sources which host AGN are removed from our analysis by means of multiple AGN indicators. We find an AGN fraction of 18 +/- 2 per cent amongst our sample and note that AGN entirely dominate at values of log [L-X/L-IR] \textgreater -3 in both hard and soft X-ray bands. From the sources which are star formation dominated, only a small fraction are individually X-ray detected and for the bulk of the sample we calculate average X-ray luminosities through stacking. We find an average soft X-ray to infrared ratio of log \textless L-SX/L-IR \textgreater = -4.3 and an average hard X-ray to infrared ratio of log \textless L-HX/L-IR \textgreater =-3.8. We report that the X-ray/IR correlation is approximately linear through the entire range of L-IR and z probed and, although broadly consistent with the local (z \textless 0.1) one, it does display some discrepancies. We suggest that these discrepancies are unlikely to be physical, i.e. due to an intrinsic change in the X-ray properties of star-forming galaxies with cosmic time, as there is no significant evidence for evolution of the L-X/L-IR ratio with redshift. Instead, they are possibly due to selection effects and remaining AGN contamination. We also examine whether dust obscuration in the galaxy plays a role in attenuating X-rays from star formation, by investigating changes in the L-X/L-IR ratio as a function of the average dust temperature. We conclude that X-rays do not suffer any measurable attenuation in the host galaxy.