Shape retrieval and alignment are a promising avenue towards turning 3D scans into lightweight CAD representations that can be used for content creation such as mobile or AR/VR gaming scenarios. Unfortunately, CAD model retrieval is limited by the availability of models in standard 3D shape collections (e.g., ShapeNet). In this work, we address this shortcoming by introducing CAD-Deform (The code for the project: https://github.com/alexeybokhovkin/CAD-Deform), a method which obtains more accurate CAD-to-scan fits by non-rigidly deforming retrieved CAD models. Our key contribution is a new non-rigid deformation model incorporating smooth transformations and preservation of sharp features, that simultaneously achieves very tight fits from CAD models to the 3D scan and maintains the clean, high-quality surface properties of hand-modeled CAD objects. A series of thorough experiments demonstrate that our method achieves significantly tighter scan-to-CAD fits, allowing a more accurate digital replica of the scanned real-world environment while preserving important geometric features present in synthetic CAD environments.