Medical images usually suffer from image degradation in clinical practice, leading to decreased performance of deep learning-based models. To resolve this problem, most previous works have focused on filtering out degradation-causing low-quality images while ignoring their potential value for models. Through effectively learning and leveraging the knowledge of degradations, models can better resist their adverse effects and avoid misdiagnosis. In this paper, we raise the problem of image quality-aware diagnosis, which aims to take advantage of low-quality images and image quality labels to achieve a more accurate and robust diagnosis. However, the diversity of degradations and superficially unrelated targets between image quality assessment and disease diagnosis makes it still quite challenging to effectively leverage quality labels to assist diagnosis. Thus, to tackle these issues, we propose a novel meta-knowledge co-embedding network, consisting of two subnets: Task Net and Meta Learner. Task Net constructs an explicit quality information utilization mechanism to enhance diagnosis via knowledge co-embedding features, while Meta Learner ensures the effectiveness and constrains the semantics of these features via meta-learning and joint-encoding masking. Superior performance on five datasets with four widely-used medical imaging modalities demonstrates the effectiveness and generalizability of our method.