Harry Leonard's three penalties were Edinburgh's only scores, as the Scottish side suffered just their second defeat at Murrayfield since February.
A cagey opening quarter of an hour saw Leonard and Priestland trade kicks deep into each other's halves but neither side could create a decent opening.
Scarlets looked most likely to make a breakthrough after 19 minutes as they shipped it wide, but Sam Hidalgo-Clyne skipped across to make a saving tackle and got to his feet quickly to win a penalty and relieve the pressure.
The Welsh side were reduced to 14 men after 23 minutes when lock Jake Ball was punished for going into the breakdown off his feet.
Edinburgh looked to capitalise immediately but a sharp break from Jack Cuthbert was halted by a knock-on and a Scarlets scrum on their own 22.
Edinburgh finally broke the deadlock after 29 minutes when a patient spell from the hosts led to Scarlets conceding a penalty on the 22, which Leonard dispatched.
However the Welshmen drew level three minutes later as Priestland's missed drop-goal was called back for a penalty after Edinburgh failed to release - the fly-half duly converted the penalty.
Scarlets survived the numerical disadvantage and grabbed the game's first try immediately after Ball's return, spinning wide for Reynolds to cross over on 35 minutes, before Priestland added the touchline conversion for a seven-point lead.
The Edinburgh pack earned a penalty on the stroke of half-time and Leonard sent the long-range effort through the posts to leave them trailing 10-6 at the break.
Leonard added another three points just two minutes into the second half after John Barclay conceded a penalty, for which he was sent to the sin bin.
Scarlets re-asserted their dominance after 46 minutes, despite being a man down, when Jordan Williams collected a box kick and sped around a crowd of players to score in the corner - but Priestland missed the conversion.
And the fly-half missed the opportunity to extend the lead further on 58 minutes - Scarlets sacked a driving Edinburgh maul and earned a penalty but Priestland narrowly missed.
WP Nel was then sent to the sin bin after 65 minutes, after one too many infringements in the scrum, to make life difficult for the hosts.
Scarlets threatened to extend their lead on 69 minutes but a knock-on at the five-metre line prevented them from capitalising on the man advantage.
But they eventually made it count eight minutes from time, when Lee Jones won a foot race to collect a through-ball but the ball came off his fingertips and Davies collected to score, with Priestland adding the two points.
Cuthbert made a late break for the hosts but he was halted by Scarlets, who held on for the victory.
|Edinburgh Rugby Score Card|
|Scarlets Score Card|