Southwest Airlines Co. benefited from an unusual deal with federal aviation regulators during the recent partial government shutdown: The carrier agreed to cover the cost of briefly recalling a furloughed safety inspector to sign off on putting three new jets into service.

The arrangement, which hasn’t been reported before, called for a $3,150 payment and three hours of administrative office time, a Federal Aviation Administration spokesman said Tuesday. He also said that senior FAA officials issued the order after agency...