It's really a slightly different structure. This has no sense of to make, to force, to oblige somebody to do something. In this case, "he has the audience eating ..." means what he is doing leads the audience to eat ... So. He tells jokes; he has them laughing. He suffers unjustified misfortune; he has them sad. He scores a goal; he has them cheering. The audience responds to him, but not with the sense of command or obligation that comes with "have sb do sth". He gets the reaction he wants by his behaviour, but not in a direct command way. They like him. So they are compliant.
Does this make sense? I hope so.