Replacement Ross Wardle's late try put the pressure on and the Dragons were piling forward for a killer blow in the dying stages, but Edinburgh held on to claim their second league win of 2014.
The hosts carved out the first opening when Taulupe Faletau's neat offload set Hallam Amos free in behind the Edinburgh line but Sam Hidalgo-Clyne forced a knock-on with his try-saving tackle.
Edinburgh continued to soak up pressure deep in their own 22 until Ross Ford was penalised in the maul, handing Jason Tovey a simple penalty after 14 minutes.
A lengthy kicking battle soon ensued but the Dragons came out on top, only for Tovey to send a miscued 25-metre drop-goal attempt into a crowd of bodies.
The away side's first real venture into Dragons territory - after the 20-minute mark - was short-lived but they came again and won a penalty after the hosts collapsed a scrum.
Cuthbert's long-range effort flew wide and he was punished shortly after the half-hour mark, when a collapsed scrum handed Tovey another straightforward penalty to double the lead.
Full-back Cuthbert hit back almost immediately, bisecting the posts with a 35-metre penalty, and Edinburgh were soon ahead.
Scott's offload sent fellow centre Andries Strauss towards the line and after he was stopped just short, Nel was on hand to pick up and drive over, Cuthbert adding the extras.
But quick thinking from Richie Rees on the stroke of half time earned the Dragons ten metres, taking them within range for Tovey and his penalty went in off the left-hand post to leave them trailing by one at the break.
Tovey's fourth penalty put the Dragons back in front within eight minutes of the restart but Scott's weaving run just two minutes later produced a second Edinburgh try.
Almost out of nothing the Scotland centre slipped through a gap 30 metres from the line and, after the TMO confirmed his grounding, Cuthbert converted.
A Cuthbert penalty on 67 minutes opened up an eight-point lead and when Lewis Evans was guilty of obstruction soon after the towering full-back split the posts once again.
But Izak van der Westhuizen was soon sent to the sin bin and the Dragons immediately capitalised, Wayne Evans' perfectly-weighted pass finding Wardle to dive over, with Tovey converting.
The Dragons' tails were up and Wardle's neat kick through piled on the pressure, eventually earning them a five-metre scrum.
A series of Dragons penalties ate up the final two minutes and, as the clock went red, one giant push from the Edinburgh scrum forced the hosts off their feet and secured the points.
|Newport Gwent Dragons Score Card|
|Edinburgh Rugby Score Card|