Multi-access edge computing (MEC) has been introduced as an enabler of low-latency performance in 4G/5G cellular networks. For the MEC-enabled cellular networks, several deployment options have been proposed by ETSI. One promising deployment option called Bump-in-the-wire does not require changes on the base station or the core network, so it has the advantage of easy deployment and low cost. However, the unchanged base station connecting to an MEC platform cannot differentiate MEC traffic from Internet traffic or prioritize it; its traffic congestion may thus cause the MEC traffic to suffer from high latency. In this work, we thus design a solution, designated PTS-MEC (Prioritized Traffic Shaping for MEC), to control the forwarding of downlink MEC/Internet traffic at the MEC and prioritize the MEC traffic based on a hierarchical MEC-prioritized fair service model. PTS-MEC alleviates the base station’s traffic congestion with a latency-aware service rate adaptor at run time by applying the service curve concept to delaying or/and skipping the Internet traffic. We prototype PTS-MEC on an open source MEC platform and evaluate it with congested cases. The evaluation result confirms the effectiveness of PTS-MEC; it can satisfy latency goals, e.g., 50 ms at the 90th percentile, within 3.70% error for MEC flows while fairly allocating remaining resource to non-MEC UEs.