Patchell double edges Scarlets past Warriors
The match was scoreless after the first quarter but Patchell burst over the line just before the half-hour mark for the game's first score.
Leigh Halfpenny added a penalty and in the second half grabbed a second three-pointer before Gareth Davies intercepted an Ali Price pass for a second try.
Replacement Peter Horne kicked a second-half penalty for the Warriors and assisted Henry Pyrgos' score, but it wasn't quite enough for Glasgow who remain at the top of Conference A.
Both of these sides are famed for their attacking rugby and they lived up to that billing in the opening quarter with play going from end to end.
Following a penalty against John Barclay for not rolling away from the tackle, Finn Russell got the chance to open the scoring with his boot but struck the post from 30 metres out.
Scarlets full-back Halfpenny had a similar opportunity to put his side three points ahead minutes later, remarkably also striking the post.
This time though Scarlets regathered off the posts and the ball eventually found its way to Patchell, who chose the most direct route possible, ricocheting through a dissipating ruck to get over the tryline.
Halfpenny missed the conversion but found his shooting boots minutes later, knocking over a long-range penalty to create an 8-0 lead.
With the last play of the half Russell kicked to the corner with a penalty awarded inside the Scarlets 22.
The Warriors couldn't make it over the line from the catch-and-drive, were awarded a penalty once again but opted for a scrum where Scarlets were able to reverse the penalty.
Horne kicked Glasgow's first points of the afternoon shortly after the restart but on 55 minutes the home side had their second try when Davies picked off a Price pass in a carbon copy of his try for Wales against Scotland in the 2018 Six Nations.
The scrumhalf touched down in the corner and Halfpenny missed the conversion but did add a penalty five minutes later.
The Warriors were in no mood to submit though and owed their first try to their replacements, with Horne breaking through a ruck into the opposition 22 and offloading to Pyrgos for the score.
At 19-8 Glasgow tried desperately for another score but it wasn't to be as Patchell jinked his way over in the corner to round off an impressive victory.
Tries:Patchell 2, G Davies
Pens: Halfpenny 3
Scarlets: 15 Leigh Halfpenny, 14 Ioan Nicholas, 13 Scott Williams, 12 Hadleigh Parkes, 11 Steff Evans, 10 Rhys Patchell, 9 Gareth Davies, 8 John Barclay, 7 James Davies, 6 Aaron Shingler, 5 Steve Cummins, 4 Tadhg Beirne, 3 Samson Lee, 2 Ken Owens (captain), 1 Rob Evans.
Replacements: 16 Ryan Elias, 17 Dylan Evans, 18 Werner Kruger, 19 Lewis Rawlins, 20 Will Boyde, 21 Aled Davies, 22 Dan Jones, 23 Steff Hughes.
Glasgow Warriors: 15 Stuart Hogg, 14 Tommy Seymour, 13 Nick Grigg, 12 Sam Johnson, 11 DTH van der Merwe, 10 Finn Russell, 9 Ali Price, 8 Ryan Wilson (captain), 7 Callum Gibbins, 6 Rob Harley, 5 Jonny Gray, 4 Tim Swinson, 3 Zander Fagerson, 2 George Turner, 1 Jamie Bhatti.
Replacements: 16 Grant Stewart, 17 Oli Kebble, 18 Siua Halanukonuka, 19 Scott Cummings, 20 Chris Fusaro, 21 Henry Pyrgos, 22 Peter Horne, 23 Lelia Masaga.
Referee: George Clancy (Ireland)
Assistant referees: David Wilkinson (Ireland), Gywn Morris (Wales)
TMO: Brian MacNeice (Ireland)