Replacement hooker Matthew Rees charged down an attempted clearance by Blair and gathered to score from close range, only for his foot to creep into touch before he grounded the ball.
A dead leg forced Priestland off and his replacement Dan Newton was wide of the target from a penalty, before Edinburgh enjoyed a concerted spell of pressure and Godman dinked over a drop goal to extend the hosts' lead to 11 points.
The Scarlets showed plenty of fighting spirit as Fenby caught past three defenders on the way to the whitewash and Newton's conversion made it a four-point game.
In the end, they had to settle for a losing bonus point as Edinburgh, 17-16 winners at Parc y Scarlets back in December, hung on to complete a season's double over the West Walians.
|Edinburgh Rugby Score Card|
|Scarlets Score Card|