A win would have put Edinburgh top of the table but in a game made difficult by constant rain the Dragons rained on their parade at Murrayfield as they went down by a single point.
Fifteen minutes into the half a big carry by Dragons number eight Grant Webb saw Edinburgh offend and Arlidge slotted a simple penalty.
Edinburgh continued to play a passing game which only led to midfield disaster and with the Dragons biding their time another penalty chance came Alridge’s way after 21 minutes and he put the Dragons 6-0 ahead.
Eight minutes later an Evans box kick had John Houston in trouble, Alan Jacobsen was the man deemed to have offended and Arlidge banged over his third penalty.
In the second-half, wing Tim Visser, number eight Ally Hogg and flanker Roddy Grant were all stopped inches short but finally the penalty was accepted and Godman slotted the points.
The game exploded into life with 14 minutes to go.
Visser got away from Aled Brew to set up the field position, the ball was shipped crossfield and Godman found Nick De Luca whose cute offload put Andy Turnbull over for the try.
Godman missed the kick and in a frantic final 12 minutes both sides had chances to win the game, Godman missing a straightforward drop goal.
With two minutes to go Toby Falatau summed up the night’s handling by spilling the ball with the line at his mercy.
Edinburgh scorers: Tries: A. Turnbull. Pens: Godman.
Newport-Gwent D’gons scorers: Pens: Arlidge 3.
EDINBURGH: S. Jones, Houston, Cairns, De Luca, Visser, Godman, Laidlaw, Jacobsen, Ford, G. Cross, S. Turnbull, MacLeod, Newlands, R. Grant, Hogg.
Replacements: A. Turnbull for Houston (41), Robertson for Cairns (32), M. Blair for Laidlaw (58), Traynor for Jacobsen (64), Kelly for Ford (58).
NEWPORT-GWENT D’GONS: Tovey, Brew, Riley, Watkins, Fussell, Arlidge, W. Evans, Gustafson, S. Jones, Palmer, Sidoli, A. Jones, MacDonald, G. Thomas, Webb.
Replacements: Gomer-Davies for Riley (69), Goodfield for S. Jones (60), Bracken for Palmer (69), Falatau for G. Thomas (72). Not Used: Harris, M. Thomas, Lee. Att: 2,066 Ref: Alan Lewis (Ireland).