Brentford were the first to threaten when Charlie MacDonald's clever flick released Sam Wood, but his poked effort was easily fielded by Shwan Jalal.
MacDonald looked the liveliest player on the pitch and he came close to breaking the deadlock in the 13th minute, when he cut in from the left and fired a low shot fractionally wide of the near post.
The home side's early pressure was eventually rewarded in the 22nd minute when Bean headed home Wood's free-kick from close range.
MacDonald was hugely unlucky not to double the lead five minutes later when he left three Bournemouth defenders for dead, only for his shot to be brilliantly pushed onto the post by Jalal.
Brentford did get their second right on the stroke of half-time, when Bean was again allowed the freedom of the six-yard box to head home Kevin O'Connor's corner.
Andy Scott's men were equally dominant in the second half and were denied a third by Jalal, whose smothering save thwarted MacDonald again.
Brentford old by Alan Connell was enduring a miserable return to Griffin Park and he wasted the visitors' best chance of the game when he scuffed his shot from six yards straight into the arms of Ben Hamer.