An early goal put the Saints in the driving seat when a through ball from Alex Darlington found Craig Jones, who crossed from the right for an unmarked Matty Williams to net from six yards out.
But the Saints' defence was dealt a double whammy inside two minutes midway through the first-half when both wing backs Danny Holmes and Chris Marriott were forced-off with hamstring injuries.
Richie Partridge's clear run to goal was halted by a foul from Paul Cochlin, who only received a yellow card, and TNS were victims of a poor decision shortly before half-time when a clearly onside Williams was flagged by an assistant as he also homed-in on goal.
And, when Scott Ruscoe crossed after going past Kevin Cooper, a Craig Jones header appeared to be clawed away from behind the goal line by keeper Craig Morris.
But TNS eventually reached the comfort zone in the 73rd-minute when Aeron Edwards was instrumental in a sweet build-up which led to the second goal, exchanging passes with Jones on the right before laying the ball off for Darlington to crash home inside the post from 15 yards.
Another flowing move saw Jones and Wood combine for Richie Partridge to slide home the third from 12 yards, but Neath did manage a consolation with virtually the last kick of the game, when Kristian O'Leary muscled his way down the byline and crossed to give Craig Hughes a simple chance.
Photo: Alex Darlington gets the congratulations after scoring the second goal.