Goals from Danny Bainbridge and Robbie Clarke saw the Wrens seemingly in command with a 2-0 half-time advantage, but Staveley hit back to level matters and take the game into extra time.
Clarke struck again within a minute of the second period of extra time to make the final scoreline 3-2 and maintain Rainworth’s 100 per cent start to the season.
The Wrens now have two home matches in the Northern Counties East League Division One – against AFC Emley on Saturday and Appleby Frodingham on Bank Holiday Monday.
Rainworth, meanwhile, have made a donation of £200 from their winning FA Cup prize fund to Dudley player Craig Shaw, who suffered a broken leg in the match.