We report the results of first-principles molecular-dynamics simulations for liquid and undercooled eutectic Au81Si19 alloys at various temperatures. Through comparisons between Au81Si19 and Au liquids, we show the strong effects of Si alloying on the atomic-scale structure, namely the occurrence of a well-defined chemical short-range order and the slowing of the formation of icosahedral local motifs as a function of temperature. Such a behavior may explain the stability of the liquid phase at the eutectic composition by an enhancement of AuSi interactions. In examining the dynamic properties of both systems, we demonstrate a strong interplay between these structural changes and the evolution of diffusivity at low temperatures. All these results yield a possible scenario for the occurrence of such an unusual deep eutectic point.