Gear skiving is an efficient and promising gear machining method, but it is mainly used to machine spur and helical gears. Curved Tooth Cylindrical Gears (CTCG) is a new type of gear characterised by a tooth trace that forms a general curve on the unfolded plane of the reference cylinder. To address the challenges of machining CTCG, a novel CTCG gear skiving method based on the new skiving tool design method and the machining motion control method was proposed. Firstly, a mathematical model of the tooth trace is established in the unfolded plane of the reference cylinder expressed as a polynomial function concerning the tooth width. Secondly, based on the existing gear skiving method, it was proposed that CTCG gear skiving be realised by continuously varying the skiving tool shaft angle during the machining process. In contrast, the skiving tool shaft angle is fixed using the standard gear skiving method. A skiving tool design method with a narrow tooth width was proposed to avoid interference caused by the changing shaft angle. Then, according to the space mesh theory and coordinate transformation, a mathematical model for solving tooth surface coordinates based on this method is established. Finally, the five-axis CNC machine tool is adopted in the virtual machining simulation software for machining simulation, and the results show that the machined tooth surface is utterly consistent with the theoretical tooth surface, which verifies the effectiveness of the proposed machining method.