Glasgow edged a tight contest at Firhill, holding their nerve as Dan Parks' haul of 14 points proved decisive in a 19-11 win.
The Warriors were looking to start their Magners League campaign with two wins from two but the Scarlets defence withstood an early Glasgow onslaught despite a couple of close things.
The first 20 minutes were notable only for two penalty misses apiece by Parks and Scarlets' Rhys Priestland, as defences cancelled each other out until a Parks break opened things up and off his pass, Kelly Brown was held inches short.
In the 31st minute an excellent Parks pass put Rob Dewey in for the opening try in the right hand corner, Stortoni having been held on the left for Glasgow to move the ball through Chris Cusiter and Al Kellock for the wing to grab his first Magners League try for Glasgow.
Parks slotted the conversion and Glasgow continued to dominate until a superb short off-load by centre Jon Davies put full-back Daniel Evans through a hole in the home defence to race away from 45 yards.
Priestland missed the conversion but as the clock wound down on the break, Glasgow indiscipline gave Priestland the chance to make amends and his first successful kick took Scarlets in at half-time 8-7 ahead.
Five minutes into the second half a beautifully weighted Parks crossfield kick found Thom Evans - only for the wing to be hauled down inches short by fellow Scot Sean Lamont.
Lamont and hooker Ken Owens took Scarlets to the Glasgow line and had the final pass not been well over Iestyn Thomas' head, a try was a certainty.
Glasgow drove downfield, Scarlets were penalised in front of heir posts and Parks restored the Glasgow lead - only for Priestland take the visitors back in front by a point with a 45-yard penalty.
Prop Deacon Manu could have stretched the Scarlets' lead had he not spilled the ball over the Glasgow line - but the home side were next on the scoreboard when Parks dropped a goal.
The Warriors started to get the edge in what had been a very even contest and with 12 minutes to go Parks brought his tally to 11 points with his second penalty and only desperate defence kept the visitors line intact.
Parks put the game beyond Scarlets with a 79th-minute penalty which denied them a bonus point which, over the course of the contest, they probably deserved.