Automatic liver tumor segmentation is of great importance for assisting doctors in liver cancer diagnosis and treatment planning. Recently, deep learning approaches trained with pixel-level annotations have contributed many breakthroughs in image segmentation. However, acquiring such accurate dense annotations is time-consuming and labor-intensive, which limits the performance of deep neural networks for medical image segmentation. We note that Couinaud segment is widely used by radiologists when recording liver cancer-related findings in the reports, since it is well-suited for describing the localization of tumors. In this paper, we propose a novel approach to train convolutional networks for liver tumor segmentation using Couinaud segment annotations. Couinaud segment annotations are image-level labels with values ranging from 1 to 8, indicating a specific region of the liver. Our proposed model, namely CouinaudNet, can estimate pseudo tumor masks from the Couinaud segment annotations as pixel-wise supervision for training a fully supervised tumor segmentation model, and it is composed of two components: 1) an inpainting network with Couinaud segment masks which can effectively remove tumors for pathological images by filling the tumor regions with plausible healthy-looking intensities; 2) a difference spotting network for segmenting the tumors, which is trained with healthy-pathological pairs generated by an effective tumor synthesis strategy. The proposed method is extensively evaluated on two liver tumor segmentation datasets. The experimental results demonstrate that our method can achieve competitive performance compared to the fully supervised counterpart and the state-of-the-art methods while requiring significantly less annotation effort.