The Welsh club were at their clinical best at Scotstoun Stadium, running in four tries as they eased out to a 28-3 lead inside 50 minutes.
Glasgow, who suffered a blow prior to kick-off when Stuart Hogg failed a late fitness test, rallied in the closing quarter, with Jonny Gray and Nick Grigg both touching down.
Yet the Warriors simply left themselves with too much ground to make up and, with the clock against them, suffered a first home defeat in the league this season.
The hosts had installed temporary seating to extend the capacity of their ground for the big occasion, yet it was the Scarlets who were sitting pretty at the break following a sensational first-half display.
What. A. Performance!
— PRO14 RUGBY (@PRO14Official) May 18, 2018
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Rhys Patchell broke the deadlock with a try he also converted, though Glasgow did respond with a penalty from the boot of Finn Russell.
However, Gareth Davies capitalised on Scott Williams’ initial break to glide over and, after Rob Evans finished a flowing counter-attacking move, the visitors were 21-3 up at the half-time interval.
Grigg saw yellow for Glasgow in the early stages of the second half, and Scarlets captain Ken Owens’ subsequent decision to reject a shot at the posts paid off when he capitalised on a rolling maul to put the Scarlets out of sight.
While the Warriors fought until the end, with Gray barging his way over the line before Grigg wriggled free to score, it is Scarlets who move on to face either Leinster or Munster at the Aviva Stadium on May 26.
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