The pressure-temperature phase diagram of the new heavy-fermion superconductor CeAu2Si2 is markedly different from those studied previously. Indeed, superconductivity emerges not on the verge but deep inside the magnetic phase, and mysteriously Tc increases with the strengthening of magnetism. In this context, we have carried out ac calorimetry, resistivity, and thermoelectric power measurements on a CeAu2Si2 single crystal under high pressure. We uncover a strong link between the enhancement of superconductivity and quantum-critical-like features in the normal-state resistivity. Non-Fermi-liquid behavior is observed around the maximum of superconductivity and enhanced scattering rates are observed close to both the emergence and the maximum of superconductivity. Furthermore we observe signatures of pressure-and temperature-driven modifications of the magnetic structure inside the antiferromagnetic phase. A comparison of the features of CeAu2Si2 and its parent compounds CeCu2Si2 and CeCu2Ge2 plotted as function of the unit-cell volume leads us to propose that critical fluctuations of a valence crossover play a crucial role in the superconducting pairing mechanism. Our study illustrates the complex interplay between magnetism, valence fluctuations, and superconductivity.