Milivojevic curled home a stunning free kick after 12 minutes and Ayew's cool finish 10 minutes later left relegation-haunted Bournemouth facing an uphill task.
The BBC, which usually only shows highlights, reached agreement to screen four top-flight games live as an unprecedented season resumes behind closed doors after a three-month stoppage because of the coronavirus pandemic.
It was not classic Saturday evening viewing for fans watching at home but Palace were much the better side as they enhanced their hopes of qualifying for Europe.
Roy Hodgson's side moved into ninth place with 42 points, four points behind fifth-placed Manchester United.
To make matters worse for Eddie Howe's side, striker Joshua King hobbled off with an ankle injury after a tackle by Milivojevic that was checked by VAR.