The second period was nervy and tense, with Paterson's 49th-minute kick leaving the match poised on a knife-edge.
The Dragons wasting several good openings to stretch their lead as both wing Fussell and full-back Dollman went close.
Edinburgh could, perhaps should, have made the hosts pay, but first Godman and then replacement David Blair were wide with drop-goals.
In the end it was down to the Dragons pack to shepherd the Welsh side home, the forwards driving patiently into Edinburgh territory to keep the ball under control and the win in the bag.
Dragons: Phillip Dollman; Gareth Wyatt, Rory Sidey, Ashley Smith, Richard Fussell; Shaun Connor, Wayne Evans; Adam Black, Tom Willis (capt), Rhys M. Thomas, Adam Jones, Luke Charteris, Joe Bearman, Richard Parks, Grant Webb. Replacements: Steve Jones, Hugh Gustafson Gustafson for Black (61), Andrew Hall A. Jones (61), Jamie Ringer Ringer for Parks (71), Rhodri Gomer Davies, Jason Tovey for Connor (54), Thomas Isaacs.
Edinburgh: Chris Paterson; Andrew Turnbull, Hugo Southwell, Nick De Luca, Mark Robertson; Phil Godman, Greig Laidlaw; Allan Jacobsen, Ross Ford, Geoff Cross, Craig Hamilton, Jim Hamilton, Matt Mustchin, Alan MacDonald, Allister Hogg (capt). Replacements: Andrew Kelly for Ford (71), Gavin Kerr, Ben Gissing Gissing for J. Hamilton (50), Scott Newlands, Ben Meyer Meyer for Laidlaw (65), David Blair for Godman (71), Jim Thompson for Robertson (49).
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