Goals from Brett Pitman and Alan Connell put the Cherries into the driving seat before substitute John Obike scored a consolation in the 86th minute.
Gavin Tomlin hit the post with a header in the 38th minute and George O'Callaghan did the same in the 62nd.
Skiverton said: "I am disappointed with the way we played and there were not many positives to take out of this. No-one likes to go out of a cup competition at the first stage to a club in a lower division.
"[Terrell] Forbes was magnificent at centre-half and [Richard] Martin handled himself well on his debut but there was not much else.
"Bournemouth played well and credit to them for playing to their strengths and working so hard. They deserved their victory and there was some fine individual performances from them. But I am just sorry we did not make it difficult for them until they were already two goals ahead."