The only goal of the game came just after the half-hour mark when Chris Brunt curled the ball in towards the far post which found Ahmed Hegazi, with the summer signing making no mistake with his header to beat a diving Asmir Begovic.
The hosts were the team on top for the majority of the match and had numerous chances to add to the scoreline including on 56 minutes when Claudio Yacob hit the side netting after finding space while taking advantage of some poor defending from Joshua King.
The Baggies never gave up on trying to extend their lead knowing a single-goal advantage is a dangerous scoreline to sit back on in the Premier League.
This included two chances in the last five minutes when first Hal Robson-Kanu did brilliantly to take down Jake Livermore's pass but then fired his shot over the bar on the turn.
Late substitute Salomon Rondon almost made a quick impact with a fine run and shot but Asmir Begovic was on hand to produce a great save to parry away the effort.
Begovic had a busy afternoon and was called upon twice in quick succession in the first half and both times to deny Jay Rodriguez. First in the tenth minute when a snap shot from the Baggies striker produce a great low save from the Cherries keeper who was then on hand three minutes later when he turned away Rodriguez who had latched onto a thrown-in.
There wasn't much to shout about for Eddie Howe's men who were lucky to escape with just the one-goal defeat while Tony Pulis and his side can be satisfied with a good performance on the opening weekend of the new season.