Southwest Airlines Co. failed to perform mandatory rudder inspections on more than 125 jets, or roughly one-fifth of its fleet, forcing it to scramble late Tuesday to negotiate an unusual arrangement with federal regulators to keep the planes flying until the checks were completed.

The drama began Tuesday afternoon, when the carrier informed the FAA that certain hydraulic inspections were missed on 128 of the company’s Boeing 737-700 models, which were voluntarily taken out of service right away. Some 80 flights were canceled...