We present measurements of the evolution of normal-galaxy X-ray emission from z~0-7 using local galaxies and galaxy samples in the ~6Ms Chandra Deep Field-South (CDF-S) survey. The majority of the CDF-S galaxies are observed at rest-frame energies above 2keV, where the emission is expected to be dominated by X-ray binary (XRB) populations; however, hot gas is expected to provide small contributions to the observed-frame <~1keV emission at z<~1. We show that a single scaling relation between X-ray luminosity (L_X_) and star-formation rate (SFR) literature, is insufficient for characterizing the average X-ray emission at all redshifts. We establish that scaling relations involving not only SFR, but also stellar mass (M_) and redshift, provide significantly improved characterizations of the average X-ray emission from normal galaxy populations at z~0-7. We further provide the first empirical constraints on the redshift evolution of X-ray emission from both low-mass XRB (LMXB) and high-mass XRB (HMXB) populations and their scalings with M_ and SFR, respectively.