The wave function in quantum cosmology is presented in the form of a rigorously defined covariant functional integral involving the relativistic gauge-breaking and ghost contributions and calculable beyond the semiclassical approximation. The functional integral is formulated both in the lorentzian and euclidean spacetimes and includes the case of the Hartle-Hawking prescription for the cosmological wave function. The relationship between this function and the on-shell effective action in quantum cosmology is established. The loop diagrammatic technique for the proposed integral is derived by setting the boundary-value problems for the gravitational and ghost-field propagators and the one-loop approximation is presented, useful for application to concrete cosmological problems.