This paper investigates the system-level throughput performance of non-orthogonal access with minimum mean squared error-based linear filtering followed by a successive interference canceller (MMSE-SIC) in the cellular uplink. Although non-orthogonal access employing the MMSE-SIC achieves the entire region of the multiuser capacity in a multiple access channel (MAC), which should be beneficial in enhancing the total user throughput and cell-edge user throughput simultaneously compared to orthogonal access, the multiplexing of multiple users within the same frequency block increases the inter-cell interference in the context of the cellular uplink. The aim of the transmission power control method investigated in this paper is to mitigate the inter-cell interference increase due to non-orthogonal user multiplexing. We employ a weighted proportional fair (PF)-based multiuser scheduling scheme to achieve a good tradeoff between the total user throughput and cell-edge user throughput. Simulation results show that non-orthogonal access employing the MMSE-SIC using the proposed transmission power control significantly enhances the system-level throughput performance compared to orthogonal access, which is widely used in 3.9 and 4G mobile communication systems.