Scottish international striker Beattie, who joined Albion from Celtic for £1.25m last month, pounced after just 23 minutes of his home debut.
While the scoreline may have been close it was a relative cruise for Albion who recorded their first win of the season after losing at Burnley on the opening day.
Bournemouth, who had started their campaign with an encouraging draw at Nottingham Forest, would have slipped to a much heavier defeat but for the heroics of goalkeeper Neil Moss.
Despite the best efforts of former England international Darren Anderton, who came closest to scoring for Bournemouth with an 87th minute cross-shot, Kevin Bond's side never really threatened to pull off a shock.
Beattie broke the deadlock in some style, latching on to a sweeping crossfield ball from Jonathan Greening before volleying his shot over Moss from the right-hand edge of the area.
Beattie was denied a second by a smart save by Moss two minutes later before the Scotsman then fired a header just over the bar in the 35th minute.
It really was one way traffic from Albion and Moss was having to work overtime to keep them at bay. He smothered a low shot from Nathan Ellington after 37 minutes but reserved his best stop of the half to deny the recalled Paul Robinson.
Robinson powered a header goalwards from Greening's corner only to see Moss's outstretched hand tip the ball over the bar.
Nathan Ellington was inches away from doubling Albion's advantage when he turned a James Morrison cross just wide in the 57th minute before debutant Bostjan Cesar flashed a header over the bar 60 seconds later.