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It was their fourth straight home win, the Dragons earning the points thanks to their three-to-one try count.
In a tight match dominated by the kickers, the Dragons did not seal their victory until seven minutes from time.
After winning a scrum against the head, home scrum-half Gareth Baber chipped ahead just outside the Edinburgh 22.
When opposite number Graeme Burns missed the ball, Baber was up to gather and score. Outside half Craig Warlow converted from the touchline and put his side clear at 34-22.
Before then Dragons’ outstanding flanker Jason Forster had grabbed tries after breaking away from mauls in the 10th and 53rd minutes and Warlow had kicked five penalty goals.
Opposite number Brendan Laney had kicked three and replacement Alistair Warnock two when Laney was off the field with blood streaming from his nose.
That was the result of a 17th-minute collision with Dragons fullback Sione Tuipulotu, who was also off briefly with a cut eye.
On a cold evening both sides tried to play expansive rugby but Newport always looked the more dangerous and might have gained a bonus point in the final minutes when Forster was held on the line as he went for his hat trick.
It was a pity a splendid display by Tuipulotu, on permit from Welsh Premier Division side Newport, was spoilt when his dropping the ball on his 22 led to Laney’s try, when he received from the resultant scrum and scythed through unopposed between the posts.