Creating high-fidelity, animatable 3D talking heads is crucial for immersive applications, yet often hindered by the prevalence of low-quality image or video sources, which yield poor 3D reconstructions. In this paper, we introduce SuperHead, a novel framework for enhancing low-resolution, animatable 3D head avatars. The core challenge lies in synthesizing high-quality geometry and textures, while ensuring both 3D and temporal consistency during animation and preserving subject identity. Despite recent progress in image, video and 3D-based super-resolution (SR), existing SR techniques are ill-equipped to handle dynamic 3D inputs. To address this, SuperHead leverages the rich priors from pre-trained 3D generative models via a novel dynamics-aware 3D inversion scheme. This process optimizes the latent representation of the generative model to produce a super-resolved 3D Gaussian Splatting (3DGS) head model, which is subsequently rigged to an underlying parametric head model (e.g., FLAME) for animation. The inversion is jointly supervised using a sparse collection of upscaled 2D face renderings and corresponding depth maps, captured from diverse facial expressions and camera viewpoints, to ensure realism under dynamic facial motions. Experiments demonstrate that SuperHead generates avatars with fine-grained facial details under dynamic motions, significantly outperforming baseline methods in visual quality.