We prove that, on a complete hyperbolic domain $D\subset \mathbb{C}^q$, any Loewner PDE associated with a Herglotz vector field of the form $H(z,t)=\Lambda(z)+O(|z|^2)$, where the eigenvalues of $\Lambda$ have strictly negative real part, admits a solution given by a family of univalent mappings $(f_t\colon D\to \mathbb{C}^q)$ which satisfies $\cup_{t\geq 0}f_t(D)=\mathbb{C}^q$. If no real resonance occurs among the eigenvalues of $\Lambda$, then the family $(e^{\Lambda t}\circ f_t)$ is uniformly bounded in a neighborhood of the origin. We also give a generalization of Pommerenke's univalence criterion on complete hyperbolic domains.