Scotland fly-half Ruaridh Jackson hobbles off holding his right hamstring
Priestland kicked two further efforts as the Welsh side went into the lead for the first time.
And Glasgow were dealt a blow in the space of a minute when flanker Harley and number eight Wilson found themselves yellow-carded, for a punch and professional foul respectively, by referee Dudley Phillips.
Then the visitors conceded a converted try deep into injury-time as Davies barged over to give the home side a 16-8 interval lead.
Glasgow showed much more urgency after the break and earned a penalty kicked by Scott Wight on 44 minutes.
The Scarlets brought on the experience of steadying influence Stephen Jones and he made an impact as both Wales stars Jonathan Davies and George North made surges towards the Glasgow line.
But a second try proved elusive as Glasgow cut the deficit to just a point with a second Wight penalty on 70 minutes.
The Scarlets pressured the Glasgow line and thought they had a try but the referee disallowed it for a knock-on leaving the home side pointless in the second half.
Newton; North; S. Williams, J. Davies; Lamont; Priestland, G. Davies; I. Thomas, M. Rees, Manu, Reed, Day, McCusker, J. Edwards, Morgan.
S. Jones for Newton (54), Iongi for Lamont (45), R. Williams for G. Davies (45), R. Jones for I. Thomas (58), Owens for M. Rees (58), R. Thomas for Manu (58), Shingler for Day (64), Turnbull for J. Edwards (76).
Dunbar for Dewey (58), Wight for Jackson (13), McConnell for Pyrgos (77), Reid for Welsh (58), Gillies for Thomson (58), Kalman for Low (28), Forrester for Fusaro (58), Beattie for R. Wilson (58), Beattie for R. Wilson (58).
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