Openside flanker Finn Fenhert gave RSA the early advantage when he powered over for the first try of the game in the opening minutes.
He carried strongly and broke several tackles before touching down near the posts, setting up Matthew Monaghan for the conversion to make it 7-0.
Both sides were evenly matched, with neither able to make the breakthrough and alter the scoreboard before half time.
Royal School Dungannon turned up the pressure early into the second half and saw an effort held up over the line by a resilient RSA defence.
Armagh then worked their way up the pitch in a few phases, which culminated in Josh Watterson offloading to Matthew Nicholl who crashed over from close range to extend the lead. Monaghan again slotted his kick to make it 14-0 to RSA.
However, RSA then had full-back James Campbell shown a yellow card, and Royal School Dungannon took advantage of their numerical supremacy.
They immediately worked the ball through the backs and it was right winger Ethan Carberry who finished well in the corner.
Dungannon then scored another in quick succession. This time the backs again combined well to send Alfie Lewis into space and he darted into the corner to finish well.
Armagh held a 14-10 lead heading into the dying minutes, and they sealed their win when the pack mauled their way over the line and captain Sam Anderson emerged with the ball.
The unsuccessful conversion signalled the end of the game, confirming Royal School Armagh as this year's Danske Bank Schools' Subsidiary Shield winners.