The 19-year-old from Portsmouth showed great skill to fire the Cherries into a 26th minute lead after he had come on as a first minute substitute for injured Josh Gowling.
Bournemouth reshuffled their pack and were under early pressure from Martin Allen's side with Lloyd Owusu and DJ Campbell both going close.
Campbell forced a great save from Gareth Stewart in the 22nd minute but it was the visitors who took a surprise lead after 26 minutes.
Keene twisted and turned showing some magical touches before firing a low shot into the net from 15 yards out.
Brentford tried to respond but the visitors kept piling on the pressure with James Hayter causing problems with his pace up front.
The striker then extended Bournemouth's lead on 41 minutes when he followed up Brian Stock's rasping drive from 25 yards that Stuart Nelson failed to hold.
Brentford almost pulled one back just before half time but James O'Connor cleared a header from a corner off the line.
Allen made two substitutions at half time but his team struggled to find a way back into the game after the restart with Keene almost extending Bournemouth's lead immediately.
Hayter almost claimed his second of the game 10 minutes later but his side-footed shot dribbled just wide.
Brentford then switched to an ambitious 3-4-3 formation in a bid to score a goal and Owusu went close but he failed to direct his header on target from substitute Sam Tillen's excellent cross.
Bournemouth hung on until the end to claim their third away win of the season.