BezierS fly-half Gonzalo Quesada kicked all 21 points for the home side to take his side to the top of Pool 1.
Neath scored the games only try, through Alfie Mocelutu, but failed to turn second-half pressure into much-needed points. The home side defended magnificently in the final minutes to deny the Welsh All Blacks a much needed win.
In a first-half kicking battle, Argentinean international Quesada, slotted penalty goals from halfway in the sixth and 10th minutes, before Shaun Connor got the visitors on the board with a penalty goal soon after.
Quesada extended Beziers' lead with a simple goal after 24 minutes before Connor levelled the scores with two fine efforts.
Butr moments later Quesada punished a high tackle from Barry Williams to
send his team into the break leading 12-9.
Italian referee Marco Salera granted more penalties in the second half, giving Quesada the opportunities to kick his fifth and sixth penalty - he duly obliged to extend the Beziers lead to 18-9.
Neath responded by scoring the first try of the match through backrow forward Alfie Mocelutu. The giant Fijian finished an excellent rolling maul by touching down.
Connor converted but missed the simple penalty-goal attempt in the 72nd minute which would have narrowed the gap to two - that proved to be the final score of the game.
Neath leave France empty handed, and with just one point now from two matches- the Welsh All Blacks have an uphill struggle to qualify.
Beziers: Pens - G Quesada 6; DGs - G Quesada.
Neath: Try - A Mocelutu; Pens - S Connor 3; Con - S Connor. Halftime: Beziers