Stoddart scored early in the second half in Edinburgh
Scarlets ended Edinburgh's run of five home Magners League wins in succession despite an error-strewn performance at a sub-zero Murrayfield.
Tries from full-back Morgan Stoddart and centre Jonathan Davies combined with 11 points from fly-half Rhys Priestland were enough for the Welsh.
Talismanic full-back Chris Paterson notched all the hosts' points courtesy of a converted try and three penalties.
However, Scarlets took the points to move up to second in the table.
It was the visitors who made the brighter start and they had an early try-scoring chance.
Edinburgh scrum-half Mike Blair, playing his first game back since Scotland's Autumn international capitulation to New Zealand, threw a terrible pass that Scarlets flanker Richie Pugh hacked ahead into the Edinburgh in-goal area.
However, Paterson, also making a return from injury, averted the danger.
Moments later, an infringement at the scrum gave Priestland the opportunity to put his side in front. He duly obliged and he then doubled the Scarlets' advantage with a second accurate kick in the 20th minute.
In contrast to his elder brother, home fly-half David Blair appeared to be at the top of his game, producing an electrifying break that yielded three points.
The move petered out, but the penalty award at the resultant ruck enabled Paterson to put his side on the board.
Classy Scarlets centre Regan King made light work of the challenging underfoot conditions as he danced his way through the home rearguard on 25 minutes.
His attempted inside offload ricocheted of the backtracking John Houston, but full-back Stoddart was unable to collect the ball as it lay inches from the line.
David Blair made another sizzling break and the hosts managed to draw level, with Paterson converting a penalty on the stroke of half-time.
The second half began with a flourish, Stoddart making amends for his earlier failure by diving over in the corner a minute after the restart.
Priestland was off the mark with the conversion attempt but stretched his side's advantage with a third successful penalty, only for Paterson to respond in similar fashion.
Centre Jonathan Davies gave a timely reminder to Wales coach Warren Gatland of his return to fitness as he capped a man-of-the-match display with a try.
He romped over having successfully fended off the home tacklers. Priestland once again converted to make it 21-9 to Nigel Davies' side.
There was further good news for Gatland as fly-half Stephen Jones made his return from a shoulder injury by coming on for the final 17 minutes.
The introduction of fresh legs appeared to spark Edinburgh into life, with replacements Greig Laidlaw and Simon Webster combining to send Webster storming over, only for the score to be chalked off after the final pass was deemed to have gone forward.
Hope of a late comeback was fuelled when Paterson crossed the whitewash to finish off a clinical move for a converted score, but the visitors withstood a late onslaught to clinch the win.
Edinburgh: Paterson, L. Jones, De Luca, Houston, Visser, D. Blair, M. Blair, Jacobsen, Ford, G. Cross, MacLeod, MacKenzie, McInally, MacDonald, Talei. Replacements: Traynor for Jacobsen (54), Kelly for Ford (57), Young for G. Cross (54), Hamilton for MacKenzie (58), Newlands for McInally (58). Not Used: Laidlaw, King, Webster.
Scarlets: Stoddart, Maule, King, J. Davies, Lamont, Priestland, Knoyle, John, M. Rees, Gardner, Cooper, Reed, Fa'amatuainu, Pugh, Lyons. Replacements: Roberts for Knoyle (63), I. Thomas for John (50), Phillips for M. Rees (50), R. Jones for Gardner (71). Not Used: Welch, Morgan, S. Jones, L. Williams.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.