The Cherries dominated the game and should have scored more than the solitary goal with James Hayter the biggest culprit when he missed a penalty.
This was Bournemouth's second 1-0 home win in a week and luck is starting to turn their way.
Bournemouth midfielder Marcus Browning hit a 20-yard drive after Steve Fletcher had nodded back but a Brentford defender threw himself in front of the effort.
Bournemouth's players appealed for a penalty claiming handball but the referee waved away the protests.
Bournemouth almost took the lead from the resulting corner with Eddie Howe getting on the end of Anderton's centre and seeing the ball roll agonisingly across the line.
Tom Pinault had Brentford's first attempt at goal in the ninth minute but his right-footed shot went harmlessly wide.
Bournemouth captain Neil Young was relieved to see Neil Moss save his intended back-pass in the 24th minute.
Bournemouth took the lead just before the break with an Anderton free-kick. Brentford centre-back David Partridge fouled Fletcher on the edge of the box and Anderton stepped up to crack the ball around the wall and into the bottom right-hand corner of the net.
Browning should have doubled Bournemouth's lead when a Brett Pitman header set him up in the 47th minute. Browning hit the shot high and wide from ten yards out and held his head in his hands.
Brentford goalkeeper Nathan Abbey pulled off a fabulous save to deny Fletcher. Pitman crossed and Fletcher glanced his header goalwards from eight yards out but the keeper made a fine one-handed save to push the ball out for a corner.
Pitman just failed to get on the end of Anderton's cross with the goal at his mercy and the chance went begging.
Hayter missed a penalty in 73rd minute awarded after he was brought down in the box. The Cherries hot-shot struck it to the left but Abbey guessed right and palmed it away.
Michael Leary should have equalised for Brentford in injury-time but he headed straight at Moss from five yards.