The Tipperary men top the Round Robin Qualifiers table on nine points after picking up their second straight win, following on from last weekend's 40-27 defeat of Monivea.
Naas winger Stuart Pringle netted an early try for the visitors this weekend but Nenagh's dominant pack were soon gaining crucial territory.
The Munster Junior League Champions struck back through scores from forwards Breen and Moylan and were 16-10 in front before Naas fly-half Joseph Moran - a former Ireland Under-18 - kicked a late penalty for a bonus point.
That leaves the Kildare men four points off the pace ahead of next week's final match-ups.
Round Robin Qualifier:
Nenagh Ormond 16 Naas 13, Nenagh
Round Robin Qualifier Fixtures:
Saturday, April 30: Naas v Monivea, Forenaughts; Nenagh Ormond v Rainey Old Boys, Nenagh, both 2.30pm