Cardiff were unable to sustain a promising first-half performance that gave them an 11-6 lead at the interval.
Chris Anderson cancelled out an early penalty goal by James Hook for Neath but fell behind again to another Hook three-pointer.
Blue & Black hopes rose when left wing Francesco Brillante raced over for his first try of the season and Anderson's second penalty goal kept the scoreboard ticking over.
It was still anyone's game in the early stages of the second half. A Price penalty converted by Hook put the visitors ahead but Cardiff hit back with another Anderson penalty goal.
Neath took a lead they were never to lose when ex-Blue & Black Dan McShane touched down and further scores came from Arwel Thomas and Gareth Morris; Hook converted two of the final three tries and kicked another penalty for a 15-point haul. Cardiff's only other score was another penalty goal from Anderson.