Newport's only highlight had been a 24th minute try for young centre Tyler Morgan with Ethan Davies converting after earlier adding a 14th minute penalty.
For Llanelli, things improved in the second half and a good off load from replacement Kirby Myhill sent prop Rhys Thomas in for his team's first try on 55 minutes. Damjanovic converted from the touchline and with Llanelli now starting to click, some good continuity play resulted in Dan Thomas touching down to extend their lead to 21-10.
Matt O'Brian responded with a penalty for Newport and five minutes later, Llanelli's No. 8 Stuart Leach received a yellow card for illegally stopping a Newport player from scoring. Referee Simon Rees had no hesitation in awarding a penalty try, which O' Brian converted to bring Newport just a point behind.
With just 14 men left to look after a one point lead, Llanelli stood up to the challenge and defended heroically.
And when man of the match Steffan Evans intercepted a pass on the halfway line, he sprinted clear to seal an impressive away win.