We study the effect of baryons on the inner dark matter profile of the first galaxies using the First Billion Years simulation between z = 16 and 6 before secular evolution sets in. Using a large statistical sample from two simulations of the same volume and cosmological initial conditions, one with and one without baryons, we are able to directly compare haloes with their baryon-free counterparts, allowing a detailed study of the modifications to the dark matter density profile due to the presence of baryons during the first billion years of galaxy formation. For each of the ≈5000 haloes in our sample (3 × 107 M⊙ ≤ Mtot ≤ 5 × 109 M⊙), we quantify the impact of the baryons using η, defined as the ratio of dark matter mass enclosed in 100 pc in the baryonic run to its counterpart without baryons. During this epoch of rapid growth of galaxies, we find that many haloes of these first galaxies show an enhancement of dark matter in the halo centre compared to the baryon-free simulation, while many others show a deficit. We find that the mean value of η is close to unity, but there is a large dispersion, with a standard deviation of 0.677. The enhancement is cyclical in time and tracks the star formation cycle of the galaxy; as gas falls to the centre and forms stars, the dark matter moves in as well. Supernova (SN) feedback then removes the gas, and the dark matter again responds to the changing potential. We study three physical models relating the motion of baryons to that of the dark matter: adiabatic contraction, dynamical friction, and rapid outflows. We find that dynamical friction plays only a very minor role, while adiabatic contraction and the rapid outflows due to feedback describe well the enhancement (or decrement) of dark matter. For haloes which show significant decrements of dark matter in the core, we find that to remove the dark matter requires an energy input between 1051 and 1053 erg. For our SN feedback proscription, this requires as a lower limit a constant star formation rate between 0.002 and 0.2 M⊙ yr-1 for the previous 5 Myr. We also find that heating due to reionization is able to prevent the formation of strong cusps for haloes which at z ˜ 12 have ≤108 M⊙. The lack of a strong cusp in these haloes remains down to z = 6, the end of our simulation.