Purpose: Blood vessels called telangiectasia are visible in skin lesions with the aid of dermoscopy. Telangiectasia are a pivotal identifying feature of basal cell carcinoma. These vessels appear thready, serpiginous, and may also appear arborizing, that is, wide vessels branch into successively thinner vessels. Due to these intricacies, their detection is not an easy task, neither with manual annotation nor with computerized techniques. In this study, we automate the segmentation of telangiectasia in dermoscopic images with a deep learning U-Net approach. Methods: We apply a combination of image processing techniques and a deep learning-based U-Net approach to detect telangiectasia in digital basal cell carcinoma skin cancer images. We compare loss functions and optimize the performance by using a combination loss function to manage class imbalance of skin versus vessel pixels. Results: We establish a baseline method for pixel-based telangiectasia detection in skin cancer lesion images. An analysis and comparison for human observer variability in annotation is also presented. Conclusion: Our approach yields Jaccard score within the variation of human observers as it addresses a new aspect of the rapidly evolving field of deep learning: automatic identification of cancer-specific structures. Further application of DL techniques to detect dermoscopic structures and handle noisy labels is warranted.