Edinburgh's impressive bonus-point win at Newport's Rodney Parade condemned the Dragons to a Heineken Cup play-off.
Dragons (11) 11
Pens: Warlow (2)
Edinburgh (16) 37
Tries: Paterson (2), Southwell, Webster
Cons: Paterson (4)
Pens: Paterson (3)
Chris Paterson and Hal Luscombe traded first-half tries, Paterson's boot giving his side a 16-11 half-time lead.
Edinburgh controlled the second 40 with tries for Hugo Southwell, Simon Webster and Paterson, the home side shut out.
The Dragons cannot now overtake the Ospreys, who are guaranteed a Heineken Cup place. The Dragons face an Italian play-off at Rodney Parade on 2 June.
That will be against the third best team in the Italian league, a game the Welsh side will be confident of winning to claim a place in the premier European tournament.
The victory moves Edinburgh up to third in the Celtic League after a solid season, but they have nothing else left to play for.
After an early exchange of penalties between Paterson and Craig Warlow, Peter Jorgensen broke the deadlock with a fine midfield break.
The Australian burst through a weak Ceri Sweeney tackle to feed Southwell, who sent Paterson in at the corner for the opening try.
Warlow immediately replied with his second penalty, and then played his part in a fine passing move along the three-quarter line that gave Luscombe the outside shoulder.
The Wales centre, making his last league appearance at Rodney Parade before a summer move to Harlequins, exploded onto Gareth Wyatt's pass and had the pace to sprint in from 50m.
Repeated Dragons infringements gave Paterson two more kickable penalties to put the visitors ahead at the break.
They extended the lead early in the second half, turnover ball spread quickly across the back line to number eight David Callam.
He cut back inside and passed to the supporting Southwell out of the tackle for the full-back to score.
Dragons captain Jason Forster - set to leave Rodney Parade in the summer - then departed with an injury, Rhys Oakley taking his place.
With the Scots in total control and encamped in the home side's half, Matt Dey cut a neat inside angle before finding Jorgensen.
The centre found Webster with an inside ball to send the Scotland wing under the posts.
The home side rallied with a series of attacks, but Webster made a thrilling break from his own try line.
He was hauled down just short, but the Gunners recycled to send man-of-the-match Paterson in for his second try, the conversion giving him 27 points in the match.
Edinburgh captain & man-of-the-match Chris Paterson:
"We love sunny days and hard pitches, they bring out the best in us.
"We're not the biggest side, but we can play some very good stuff.
"Credit to the Dragons, they allowed us to play and we've had some good games with them.
"We've come through some tough times at Edinburgh, but now we're starting to get some away form to go with results at home."
Dragons captain Jason Forster:
"It's very disappointing - I've said we're a Jekyll and Hyde side but that sort of performance isn't acceptable.
"We have to take a long, hard look at ourselves, the fans deserve better than that.
"It's been a very disappointing end to the season and there are some trying times ahead."
Dragons forwards coach Leigh Jones:
"We have to own up to that loss - we under-estimated Edinburgh and thought they might roll over, but we were the ones to do that.
"We have to get back on track against Borders next to get ourselves ready for the Italians."
Dragons: Wyatt, Hunt, Luscombe, Sweeney, Brew, Warlow, Baber, Black, Brown, R. Thomas, Gough, Charteris, Ringer, Owen, Forster.
Replacements: Bryant for Sweeney (75), Ireland for Baber (54), Robinson for R. Thomas (61), Hall for Gough (66), Oakley for Forster (47). Not Used: Anthony, A. Thomas.
Edinburgh: Southwell, Paterson, Jorgensen, Dey, Webster, Monro, Lawson, Jacobsen, Kelly, Dickinson, Kellock, Murray, Mustchin, Callam, Strokosch.
Replacements: Godman for Monro (73), Blair for Lawson (73), Smith for Jacobsen (63), Rennie for Kelly (76), Pringle for Kellock (76). Not Used: Traynor, Leonelli Morey.
Referee: Donal Courtney (Ireland)