With Priestland pulling the strings expertly from fly half, the Scarlets ran in three tries with wingers Nic Reynolds and Jordan Williams scoring along with Jonathan Davies
Edinburgh fly-half Harry Leonard opened the scoring with a penalty but that was quickly cancelled out by Priestland.
Reynolds dived over for the game’s opening try and Priestland then fired over an excellent conversion from the sideline to put the Scarlets 10-3 ahead, reduced by three points after a penalty by Leonard on half-time.
Williams startled the home defence to touch down in the corner before Davies finished Edinburgh off with a clinical finish.
Scarlets head coach Simon Easterby said he thought his side managed the game well with an accurate and pretty effective display.
“I think we came a little unstuck on occasions in the contact area but on the whole we were good and securing that result was exactly what we needed after last weekend,” he said referring to the defeat to the Dragons.”
Easterby added: “There were good individual performances but what we saw was a huge effort from everyone. It was really important for us to step up against Edinburgh. It is a long season and nobody was panicking after that result last weekend. It wasn’t good enough, we didn’t perform well enough but we thought we could have come away with something from Rodney Parade but we didn’t.
“We have to build on this performance now. This is the standard and the benchmark to adhere to. We have Glasgow at home next week and they have been going so well – so it’s a massive game for us to build consistency before we head into Europe.”
