Three-dimensional mesh post-processing is an important task because low-precision hardware and a poor capture environment will inevitably lead to unordered point clouds with unwanted noise and holes that should be suitably corrected while preserving the original shapes and details. Although many 3D mesh data-processing approaches have been proposed over several decades, the resulting 3D mesh often has artifacts that must be removed and loses important original details that should otherwise be maintained. To address these issues, we propose a novel 3D mesh completion and denoising system with a deep learning framework that reconstructs a high-quality mesh structure from input mesh data with several holes and various types of noise. We build upon SpiralNet by using a variational deep autoencoder with anisotropic filters that apply different convolutional filters to each vertex of the 3D mesh. Experimental results show that the proposed method enhances the reconstruction quality and achieves better accuracy compared to previous neural network systems.