I agree with msbella that the song isn't completely jazz but the big band arrangement swings, pretty wildly. At one point, two quick drum hits in time as Gilda moves her hips are a nod to striptease shows. The close-up where she fluffs her hair seductively is something Marilyn Monroe copied several years later, consciously or unconsciously.
It's also curious what we don't see in the scene. Johnny is shown in only one quick reaction shot. He isn't pleased, obviously, but he's enough of a showman, and has enough sense, to let the show go on. While the music is playing, he can still tell hims