The fixture had been transferred from Llangennech RFC due to the freezing weather conditions, the pitch at Ospreys HQ passed playable earlier in the day.
The Eyasses opened the scoring with a try from wing Jordan Adderley converted by Luke Price before the Scarlets outside half Daniel Jones reduced the deficit with a penalty.
The hosts looked likely to go in at the break holding a 7-3 lead, only for Scarlets centre Elgan Wilson to cross for a try on the stroke of half time converted by Daniel Jones, giving their team a 10-7 advantage at the whistle.
The second half produced a number of scoring opportunities for both sides to take control of the game but it was a second unconverted try from Eyasses wing Jordan Adderley that proved the decisive score mid way through the second half.
Continuous pressure from the Scarlets in the final minutes with the Eyasses reduced to 14 men came to nothing as the Young Ospreys defence held firm, ensuring they reached their fifth final in eight seasons of the competition.
