Oil spills pose severe environmental risks, making early detection crucial for effective response and mitigation. As Synthetic Aperture Radar (SAR) images operate under all-weather conditions, SAR-based oil spill segmentation enables fast and robust monitoring. However, when using deep learning models, SAR oil spill segmentation often struggles in training due to the scarcity of labeled data. To address this limitation, we propose a diffusion-based data augmentation with knowledge transfer (DAKTer) strategy. Our DAKTer strategy enables a diffusion model to generate SAR oil spill images along with soft label pairs, which offer richer class probability distributions than segmentation masks (i.e. hard labels). Also, for reliable joint generation of high-quality SAR images and well-aligned soft labels, we introduce an SNR-based balancing factor aligning the noise corruption process of both modalilties in diffusion models. By leveraging the generated SAR images and soft labels, a student segmentation model can learn robust feature representations without teacher models trained for the same task, improving its ability to segment oil spill regions. Extensive experiments demonstrate that our DAKTer strategy effectively transfers the knowledge of per-pixel class probabilities to the student segmentation model to distinguish the oil spill regions from other look-alike regions in the SAR images. Our DAKTer strategy boosts various segmentation models to achieve superior performance with large margins compared to other generative data augmentation methods.