Oliver, who is currently controller of English at the braodcaster's Global News division, is a former BBC News at Ten editor and ran the broadcaster's general election campaign coverage last year.
Coulson, previously editor of the News of the World, resigned from the communications post last month, claiming that "continued coverage" of the phone-hacking scandal at the tabloid was making it difficult to do his job.
According to a report by BBC News, Oliver, who will be employed by the government as a special advisor, was approached late last week by Coulson and will receive a salary of £140,000.
"I'm delighted to be joining David Cameron and his team at such an exciting and challenging time," he said in a statement.
"It's difficult to leave the BBC after a fascinating few years - but this is an opportunity I can't turn down."
Cameron said in a statement yesterday that he was "very pleased" by the appointment.
"Craig has formidable experience as a broadcast journalist. He will do an excellent job in explaining and communicating the government's programme."
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