In this paper, a new congestion control mechanism for flow-aware networks, that is, enhanced flushing mechanism with priority (EFMP) is introduced. Proposed by J. Roberts and S. Oueslati in 2004, flow-aware networking is a concept that ensures quality of service in networks based on flows. EFMP improves the efficiency of transmission in FAN by introducing new policies in the admission control block of a cross-protect router. The mechanism is thoroughly described, analysed, and validated through simulations. The proposed mechanism inherits the advantages of previously known admission control schemes while ensuring fast acceptance times of new streaming flows and proper transmission of the remaining traffic. Moreover, the comparison with the multilayer flow-aware networks architecture shows that EFMP ensures better performance and lower cost of transmission in a network with limited resources.