Priestland added another two kicks without reply shortly afterwards to extend the Scarlets' lead to 13 points, before Nick Evans finally got the hosts on the scoreboard with a penalty of his own.
Harlequins began to crank up the pressure towards the end of the first half and were rewarded with their first try of the match courtesy of Mike Brown.
The Scarlets would end the half on top, however, with Scott Williams going over with just one minute to play to extend the Welsh side's lead to 10 points going into the break.
Samson Lee was sent to the sin bin at the start of the second half and Quins immediately looked to take advantage, closing the gap through an Evans drop goal.
Two more penalties for either side followed before Jordan Williams danced his way through the hosts' defence for his side's third try of the day.
Harlequins' hopes of pulling off a late comeback were raised by Brown when he went over in the corner for his second try of the match, before Evans slotted a difficult kick to make it 26-33 with 10 minutes remaining.
The visitors missed a chance to move more than a point ahead of Harlequins when they dragged a penalty wide with six minutes left, but the hosts were not able to take advantage in the closing stages as the Scarlets held out for the win.