We use deep HST STIS and NICMOS images of three spectroscopically confirmed galaxy counterparts of high-redshift damped Lyα (DLA) absorbers (one of which is a new discovery) to test the hypothesis that high-redshift DLA galaxies are Lyman break galaxies. If this hypothesis is correct, the emission properties of DLA galaxies must lie within the range of emission properties measured for Lyman break galaxies of similar absolute magnitude. This will be true regardless of selection biases in the sample of detected DLA galaxies. We test this prediction using several emission properties: half-light radius, radial profile (Sérsic n-parameter), optical to near-infrared color, morphology, Lyα emission equivalent width, and Lyα emission velocity structure. In all cases the measured values for the DLA galaxies lie within the range measured for the population of Lyman break galaxies. None of the measurements is in conflict with the prediction. We conclude that the measured emission properties of the three DLA galaxies studied here are consistent with the conjecture that high-redshift DLA galaxies are Lyman break galaxies. We show that this result does not conflict with the observation that the few high-redshift DLA galaxies discovered are mostly fainter than spectroscopically confirmed L* Lyman break galaxies.