Wednesday manager Gary Megson made 10 changes from the team which overcame Blackpool in the Carling Cup on Thursday night, a match which went to extra time and penalties.
So he can have no excuse for their first defeat of the season as a youthful Bournemouth took control after a shakey start.
Barrett, signed from Crystal Palace, put his team in front in the 38th minute with a header at the far post from Marc Pugh's corner.
Jose Semedo and Gary Nadine each should have levelled but headed wide from close range before Bournemouth made sure of the points in the 84th minute.
Substitute Arter finished off good work by Lyle Taylor and Liam Feeney by drilling a left-foot shot past goalkeeper Nicky Weaver.