We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point-splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g. unbounded evolution operators). Nevertheless, applying new mathematical methods, we show existence of maximal/global (in time) solutions to the SCE for vacuum-like states, and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.