Reigning champions the Ospreys know they have to secure a victory when they play Italian side Benetton Treviso at the Liberty Stadium on Saturday to keep themselves in the race for a place in the Pro12 top four.
The Warriors came into this game on the back of a confidence-boosting 51-24 win over Munster in their last Pro12 outing, but they did not come close to troubling a Scarlets side who recorded their fifth straight home win in all competitions.
The Welsh region was still reeling from the news that wing George North had agreed a three-year deal to join Northampton Saints earlier in the week.
North had found himself at the centre of the recent wrangling between the Welsh Rugby Union and Regional Rugby Wales and was probably happy to just be playing again.
But it was fly-half Williams, another departing Scarlets, who really caught the eye with some impressive kicking which saw him slot over four penalties, two conversions and a drop goal for his side
The 21-year-old, who leaves at the end of the season for the Leicester Tigers, slotted over two early penalties to give the home side a 6-0 lead midway through the first half that they would never surrender.
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It was hooker Owens, who replaced Matthew Rees in the starting line-up, who made the Scarlets domination really tell when he drove over from short range after the home side had exerted real pressure on the Scottish side's line.
Owen Williams added the conversion before Glasgow managed a response with a penalty by fly-half Ruaridh Jackson to make it 13-3. The Scarlets fly-half replied with an opportunistic drop goal before his opposite number for Glasgow added a penalty to make it 16-6 at half-time.
Former Wales Under-20s half-back Williams missed his third penalty before wing North's deft pass put namesake and full-back Liam Williams over for the Scarlets second try at the start of the second half. Owen Williams added the conversion and a third penalty to give the home side a 20-point cushion midway through the half.
Glasgow replacement fly-half Duncan Weir missed a penalty chance to haul the visitors back into the contest before Owen Williams struck again for the Scarlets with his fourth penalty to secure a vital win.
The Scarlets host Cardiff Blues and Benetton Treviso in their remaining league matches, while rivals the Ospreys have a much tougher run-in with visits to Glasgow and Leinster to come.
: Liam Williams; George North, Jonathan Davies, Scott Williams, Andy Fenby; Owen Williams, Aled Davies, Phil John, Ken Owens, Samson Lee, George Earle, Johan Synman, Aaron Shingler, Rob McCusker (capt), Johnathan Edwards.
: Stuart Hogg, Sean Maitland, Mark Bennett, Peter Horne, DTH van der Merwe, Ruaridh Jackson, Henry Pyrgos, Gordon Reid, Pat MacArthur, Moray Low, Tom Ryder, Al Kellock (Captain), Josh Strauss, John Barclay, Ryan Wilson.
Dougie Hall, Ryan Grant, Ed Kalman, Tim Swinson, Rob Harley, Niko Matawalu, Duncan Weir, Sean Lamont.
George Clancy (IRFU, 53rd competition game)
Neil Hennessy, Jon Mason (both WRU)
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