In this article, we propose a non-orthogonal resource scheduling (NORS) scheme with an enhanced preamble detection (PD) method for cellular random access (RA) systems. The enhanced PD method is designed based on a deep learning technique that can classify three PD statuses: idle, collision-free, and collision. Because of this collision resolution capability, the base station (BS) can exploit radio resources non-orthogonally in contention-based RA protocols. In addition, we mathematically analyze the performance of our proposed NORS scheme in terms of RA success probability and resource utilization efficiency and conduct a comparison with the baseline orthogonal resource scheduling (ORS) scheme. Through simulations, we verify the validity of our mathematical analysis. As a result, our results will be a useful guideline for resource scheduling and provide support for massive connectivity.