Neath's fly half James Thomas opened the scoring after six minutes with a penalty before three Richard Powell penalties after 10, 20 and 23 minutes put Bedwas in charge.
A well worked try by Neath lock Leon Ward reduced the deficit after 29 minutes and five minutes later Thomas added a second penalty.
Neath stretched their lead after 37 minutes after winning quick ball from a line out scrum half where Tom James darted through a huge gap in the Bedwas defence to cross for a try, James Thomas converting.
Richard Powell had the last say of the half with half with a penalty to reduce arrears to six points.
The second half took a while to get going before Thomas extended Neath's lead with a penalty.
Bedwas rung the changes after 58 minutes and immediately reduced the arrears with a great finish by Rory Budina, Powell converting. Thomas added a fourth penalty after Bedwas replacement No 8 Dan Crimmins was sin binned for a ruck infringement.
Bedwas fought hard to get back into the game and with the clock running down won a five metre scrum. Neath replacement prop Roy Jones was taken off with an injury and with no replacement front row players on the bench passive scrums were called.
Neath won the scrum and booted the ball into touch to claim the win to howls of derision from the Bedwas faithful.
Elsewhere in the Premiership Llandovery were held to a 10-10 draw by Aberavon to record their third stalemate of the campaign.
In their last home league game of the season the Drovers were fortunate to salvage a draw after the Wizards' equalising try came in the fourth minute of stoppage time through full back Richard Carter. A minute later Carter missed a long range penalty to win the game.
Cross Keys' hopes of a play-off spot suffered a setback when visitors Carmarthen Quins triumphed 3-7 in a low scoring affair at Pandy Park.
Dorian Jones opened the scoring for the home side before Quins centre Rhys Williams scored the only try of the match in the 61st minute. Craig Evans added the conversion and the Quins had to withstand a late challenge before confirming their second place in the league.
Despite putting up another spirited display, Swansea were pipped at the post 13-14 by Bridgend while Cardiff succumbed 24-31 to Llanelli.
Principality Premiership results
Bedwas 19 - 24 Neath
Cardiff 24 - 31 Llanelli
Cross Keys 3 - 7 Carmarthen
Llandovery 10 - 10 Aberavon
Swansea 13 - 14 Bridgend