As machine learning becomes increasingly prevalent in critical fields such as healthcare, ensuring the safety and reliability of machine learning systems becomes paramount. A key component of reliability is the ability to estimate uncertainty, which enables the identification of areas of high and low confidence and helps to minimize the risk of error. In this study, we propose a machine learning pipeline called U-PASS tailored for clinical applications that incorporates uncertainty estimation at every stage of the process, including data acquisition, training, and model deployment. The training process is divided into a supervised pre-training step and a semi-supervised finetuning step. We apply our uncertainty-guided deep learning pipeline to the challenging problem of sleep staging and demonstrate that it systematically improves performance at every stage. By optimizing the training dataset, actively seeking informative samples, and deferring the most uncertain samples to an expert, we achieve an expert-level accuracy of 85% on a challenging clinical dataset of elderly sleep apnea patients, representing a significant improvement over the baseline accuracy of 75%. U-PASS represents a promising approach to incorporating uncertainty estimation into machine learning pipelines, thereby improving their reliability and unlocking their potential in clinical settings.