Newport opened up an 18-3 gap at half-time through tries from Joe Bedford and Aled Brew. Jason Tovey added two penalties and a conversion with Phil Godman responding for the visitors.
Tovey continued his fine form with the boot in the second half with another three-pointer before Grant Gilchrist offered the visitors a glimmer of hope.
It's been a tumultuous domestic campaign for both of these sides. Despite qualifying for the Heineken Cup quarter-finals, Edinburgh have only won four league games this season.
Meanwhile, the Dragons have gone backwards from last year where they showed significant improvement, and will lose two of their better players at the end of the season in Luke Charteris and Brew.
It was the visitors who opened the scoring through Godman before Gavin Thomas was sin-binned for Newport.
However, they showed impressive resilience to fight back and Tovey added successive penalties with Thomas off the field.
That seemed to spur the hosts on and, after Sean Cox was sin-binned for the Scottish outfit, Bedford went over before Brew added a second try as the Dragons led 18-3 at the break.
The second half continued to be lively with both sides looking to move the ball, but it wasn't until the 61st minute when Tovey struck his third penalty.
But Edinburgh responded when Gilchrist barrelled across the whitewash, Gregor Hunter converting, but Newport comfortably held on.