In the present mainstream cosmology, matter and space-time emerged from a singularity and evolved through four distinct periods: early inflation, radiation, dark matter, and late-time inflation (driven by dark energy). During the radiation and dark matter dominated stages, the universe is decelerating while the early and late-time inflations are accelerating stages. A possible connection between the accelerating periods remains unknown, and, even more intriguing, the best dark energy candidate powering the present accelerating stage ( Λ -vacuum) is plagued with the cosmological constant and coincidence puzzles. Here we propose an alternative solution for such problems based on a large class of time-dependent vacuum energy density models in the form of power series of the Hubble rate, Λ = Λ ( H ) . The proposed class of Λ ( H ) -decaying vacuum model provides: (i) a new mechanism for inflation (different from the usual inflaton models), (ii) a natural mechanism for a graceful exit, which is universal for the whole class of models; (iii) the currently accelerated expansion of the universe, (iv) a mild dynamical dark energy at present; and (v) a final de Sitter stage. Remarkably, the late-time cosmic expansion history of our class of models is very close to the concordance Λ CDM model, but above all it furnishes the necessary smooth link between the initial and final de Sitter stages through the radiation- and matter-dominated epochs.