In this paper, a Rate-Splitting Multiple Access (RSMA) scheme is proposed to assist a Mobile Edge Computing (MEC) system where local computation tasks from two users are offloaded to the MEC server, facilitated by uplink RSMA for processing. The efficiency of the MEC service is hence primarily influenced by the RSMA-aided task offloading phase and the subsequent task computation phase, where reliable and low-latency communication is required. For this practical consideration, short-packet communication in the Finite Blocklength (FBL) regime is introduced. In this context, we propose a novel uplink RSMA-aided MEC framework and derive the overall Successful Computation Probability (SCP) with FBL consideration. To maximize the SCP of our proposed RSMA-aided MEC, we strategically optimize: (1) the task offloading factor which determines the number of tasks to be offloaded and processed by the MEC server; (2) the transmit power allocation between different RSMA streams; and (3) the task-splitting factor which decides how many tasks are allocated to splitting streams, while adhering to FBL constraints. To address the strong coupling between these variables in the SCP expression, we apply the Alternative Optimization method, which formulates tractable subproblems to optimize each variable iteratively. The resultant non-convex subproblems are then tackled by Successive Convex Approximation. Numerical results demonstrate that applying uplink RSMA in the MEC system with FBL constraints can not only improve the SCP performance but also provide lower latency in comparison to conventional transmission scheme such as Non-orthogonal Multiple Access (NOMA).