We present a model for the Universe in which quantum anomalies are argued to play an important dual role: they are responsible for generating matter-antimatter asymmetry, but also provide time-dependent contributions to the vacuum energy density of running-vacuum type, which drive the Universe evolution. During the inflationary phase of a string-inspired Universe, and its subsequent exit, the existence of primordial gravitational waves induce gravitational anomalies, which couple to the Kalb-Ramond (KR)axion field of the massless gravitational string multiplet. Such anomalous CP violating interactions have two important effects: first, they lead to contributions to the vacuum energy density of the form appearing in the running vacuum model (RVM) framework, proportional to both the square and the fourth power of the effective Hubble parameter $H$, the $H^4$ terms being associated with anomaly-induced inflation within the RVM framework. Second, there is an undiluted KR axion at the end of inflation, which plays an important role in generating matter-antimatter asymmetry through baryogenesis via leptogenesis in models with heavy right handed neutrinos. As the Universe exits inflation and enters a radiation dominated era, the generation of chiral fermionic matter is responsible for the cancellation of gravitational anomalies, thus restoring diffeomorphism invariance for the matter/radiation (quantum) theory, as required for consistency. Chiral U(1) anomalies may remain uncompensated, though, during matter/radiation dominance, providing RVM-like $H^2$ and $H^4$ contributions to the Universe energy density. Finally, in the current era, gravitational anomalies resurface, leading to much weaker RVM-like $H_0^2$ contributions to the vacuum energy density.Our model favours axionic Dark Matter,the source of which can be the KR axion.
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