Existing potential functions (PFs) utilized in autonomous vehicles mainly focus on solving the path-planning problems in some conventional driving scenarios; thus, their performance may not be satisfactory in the context of emergency obstacle avoidance. Therefore, we propose a novel model predictive path-planning controller (MPPC) combined with PFs to handle complex traffic scenarios (e.g., emergency avoidance when a sudden accident occurs). Specifically, to enhance the safety of the PFs, we developed an MPPC to handle an emergency case with a sigmoid-based safe passage embedded in the MPC constraints (SPMPC) with a specific triggering analysis algorithm on monitoring traffic emergencies. The presented PF-SPMPC algorithm was compiled in a comparative simulation study using MATLAB/Simulink and CarSim. The algorithm outperformed the latest PF-MPC approach to eliminate the severe tire oscillations and guarantee autonomous driving safety when handling the traffic emergency avoidance scenario.