A resilient Scarlets side snatched a valuable win 29-28 and four points which takes them to 25 points behind leader's Ulster's 24 - a third of the way to get to the 60-point bracket that's an average for a top four play-off position in the RaboDirect PRO12.
Despite a yellow card for scrum-half Gareth Davies, which affected the momentum of the game in the second half, in the end the determination and spirit of the Scarlets with the impact of the subs at the end of the game made the difference in holding out a win.
Scarlets Attack Coach Mark Jones said: "A tough and hard-fought game and one we are pleased to win; particularly as this is a tough place to go and historically we haven't always performed away in Scotland.
"After a tough fortnight, the boys really showed their commitment and determination and their mental strength to come away and get a win tonight. We should not underestimate how important a match this was in the context of the Scarlets season. Edinburgh were hurting after their poor European results so we knew we could expect a lot of intensity and intent from the home side.
"Our focus was to get to back on track and back up our results so far this season in the league and it is a big boost for us after the disappointment of the last fortnight. I am very proud of the boys determination and spirit tonight; we've talked about closing out games and particularly being able to finish teams off as we did under pressure tonight. This is probably a very significant result away from home.
"We look forward to our home game against Zebre next Saturday and hopefully continuing a strong run in the RaboDirect PRO12."
The Welshmen have already won in Scotland this season: 18-13 at Glasgow on 7 September.
Each team won its respective home game when the two sides met last season - this is the only the Scarlets' second win at Murrayfield in 10 visits so far in the competition's history.
Mark Jones added: "We have some decent play from set-piece ball and to secure three tries form set play was good to see. The boys worked hard tonight across the field and showed great mental strength after the last two games in Europe."
Jones complimented the input from some of the Scarlets new faces on the field in Scotland including Argentinian second row Tomas Vallejos and a welcome to the Scarlets shirt for former Osprey Gareth Owen who came on for Liam Williams and showed plenty of adventure and pace during an exciting break in the second half.
"We still had some handling errors in our game in attack and that hampered our flow but overall the boys worked hard for that win and we are all just pleased to put the points on the board ahead of our next league fixture.
"We lose a lot of our players to Wales now (12 in total) but we need to keep a good momentum in our season and we have a lot of talent, young players who will step up and play for the Scarlets jersey."