Although Glynneath started off the brighter with two penalties from full back David Price, Rumney got the opening try of the game when flanker Daniel Morris, outstanding throughout along side No 8 Lewis Mills, crashed through for fly-half Andrew Carlson to convert.
With prop John Leadbeater shining in tight and loose, Rumney went further ahead when full-back Owen Tresilian added a good penalty for a 10-6 interval lead.
Glynneath responded quickly after half time with a long distance try by wing Ben Powell from a vital turnover to go into the narrowest of leads.
With captain and flanker Dan Lewis and lock Jon Griffiths working hard, Glynneath exerted increasing pressure and went further ahead after a penalty by centre Greg Roberts.
Raising their game in the closing minutes Rumney came close to a try on two occasions only to be kept out by a hard tackling defence.
Try D Morris
Con A Carlson
Pen O Tresilian
Try B Powell
Pens D Price (2), G Roberts