The Reds went into the game on the back of a six-match winning streak, while the Eagles were without an away win in five attempts, and were quoted a 5/1 shot by bookies to end their drought.
And their cause worsened on 40 minutes when Paul Fowler received a straight red from referee Huw Jones for a two-footed challenge on Ashley Evans.
But, three minutes later, Matty Collins scored his first goal of the season to hand Neath the lead, firing a superb 20-yarder after a moment of magic from Lee Trundle.
Stuart Jones and Chris Venables wasted good chances for Llanelli and Kris Thomas fired just wide as the hosts dominated the second period.
Home player-boss Andy Legg sacrificed defenders and midfielders to bring on strikers Jordan Follows, Craig Moses and Chris Llewellyn as the game wore on, but the hosts were unable to make the breakthrough against some resolute defending from the visitors.