In a tight and edgy first half, a slick score from Rosser gave Dragons a slender 10-5 lead at the interval.
Rosser grabbed his of the second game after the break but a try from Edinburgh’s Darcy Graham set up a grandstand finish as Dragons held firm late on to secure a third league win of the season.
Dragons made five changes from the last league outing with Wales ace Tyler Morgan back involved, while Leon Brown was named on the bench.
Rynard Landman, Ollie Griffiths and Ryan Bevington also came into the pack while Rosser returned on the wing for a big encounter.
Griffiths and Landman were immediately into the action with strong runs while scrum half Rhodri Williams made a half break through tackles to try and give Dragons early go forward.
Back row Griffiths was showing an incredible work-rate in the opening exchanges as he raced through another gap and tore into the visiting 22 to huge cheers.
Centre Jarryd Sage then twice ook the ball into contact at pace before Edinburgh infringed on the floor and Jason Tovey slotted the penalty.
Edinburgh showed their counter attacking menace as wing Duhan van der Merwe used pace and quick thinking to kick in behind, but the alert Rhodri Williams was back to mop up danger and clear.
Both teams were looking to take control as temperatures plummeted, with the all-action Griffiths securing another turnover.
The back three then linked to devastating effect as Amos brilliantly took a high ball, full back Jordan Williams exploited a gap to wriggle out of a tackle and fed Rosser to blast down the touchline. Tovey converted for a 10-0 lead.
Edinburgh looked to hit back immediately as they broke up the middle through a combination of Luke Hamilton and Callum Hunter-Hill.
The Scottish side then won a series of penalties and inched ever closer to the try line, as they repeatedly took scrums instead of the easy three points in the shadow of the posts.
Referee Barry consulted the TMO for a possible try for Edinburgh centre Chris Dean, but ruled it out for a double movement on the line.
However, the visitors were rewarded for a dominant spell on the stroke of half-time as Dean broke away in midfield and fired a long pass out to Dougie Fife to score out wide.
Dragons strived to wrestle back control at the start of the second half, with Rhodri Williams neatly grubber kicking in behind and almost finding a flying Rosser.
Tovey then saw a penalty pull up just short of the target, after a trademark barreling run from Nic Cudd saw Edinburgh concede a shot at the posts.
A stolen lineout then saw Harri Keddie fling a long pass to Griffiths who blasted a huge hole in the Scottish defence and fed the marauding Amos on his shoulder.
A second home try looked imminent and duly arrived as Rhodri Williams made a sniping break before Tovey floated a long pass to Jordan Williams who, once again, fed Rosser for his second try.
Tovey extended the lead with a penalty ust after the hour, after a huge home scrum, but back came the visitors to set up a nerve-shredding finale.
A try from Graham, after a lengthy consultation with the TMO, put Edinburgh within six points going into the final quarter.
Tovey saw a penalty just drift wide with six minutes left, after Edinburgh were penalised for pulling down a driving maul.
And a breathless finish saw Dragons repel 25 phases of intense pressure before stealing the ball to signal the final whistle to a huge ovation.
10m Tovey p 3-0
25m Rosser t Tovey c 10-0
40m Fife t 10-5
55m Rosser t 15-5
65m Tovey p 18-5
68m Graham t van der Walt c 18-12
Referee: Stuart Barry