Scarlets surrendered an early 8-0 lead before replacement fly-half Owen Williams kicked a penalty 13 minutes from time which was enough to condemn Edinburgh to an 11 straight defeat in all competitions.
Llanelli had Wales squad member Liam Williams at full-back while Edinburgh had Ben Cairn at centre replacing Nick De Luca who received a 13-week ban for a tip tackle in the defeat at Ospreys last week.
It started well enough for Scarlets as they opened up that eight-point lead only to see it erased by 10 points in four minutes from Edinburgh.
Scarlets opened the scoring on nine minutes with a penalty for outside-half Aled Thomas.
And the home lead was increased five minutes last as Thomas sent wing Andy Fenby away for a break and the opening try.
Fenby was given a free run to the line after full-back Greig Tonks missed his cover tackle.
From 8-0 down Edinburgh needed to do something and immediately went up the other end and earned a penalty which Harry Leonard kicked from the 22.
That obviously gave the Scots some confidence as they cruised into the lead when Leonard's long pass opened the way for wing Dougie Fife to score.
Leonard's conversion put Edinburgh ahead, a two-point advantage that they took into half-time.
Edinburgh won the first battle of the second half when lively scrum-half Richie Rees charged down a kick in the 22. From that position Edinburgh earned a penalty which Leonard kicked to extend their lead to 13-8.
Scarlets battered their way into the Edinburgh 22 through their forwards and eventually earned a penalty which Thomas kicked to reduce the deficit to two points.
Owen Williams replaced Thomas at outside-half and had a penalty shot to regain the lead but his attempt rebound off an upright.
Scarlets regained possession and got over the line through flanker Johnathan Edwards but the TMO did not award the try.
And another chance to take the lead went begging when Owen Williams missed a second penalty attempt on 59 minutes.
Fife was sent to the sin-bin on the hour mark for a professional foul but from the resulting kick Williams made it three misses on the trot.
It took an easy attempt in front of the posts, after Edinburgh were adjudged offside, for Williams to break his duck and claim the lead off the visitors on 67 minutes, sealing Scarlets' victory in the process.
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- Liam Williams