Kenny Logan shook off the effects of an ankle injury to score two tries as Glasgow downed the Scarlets.
Glasgow (37) 40
Tries: Beveridge, Logan 2, Howarth
Cons: Howarth 4
Pens: Howarth 4
Scarlets (10) 29
Tries: B Davies, Finau, Phillips, G Evans.
Cons: Bowen 3
But the Scots' triumph was tarnished when prop Euan Murray was red-carded for illegal use of the knee.
Graeme Beveridge and Calvin Howarth, whose kicking provided Glasgow's other scoring, grabbed the other tries.
Barry Davies, Salesi Finau, Mark Phillips and Garan Evans were Scarlets' try scorers, with kicker Gareth Bowen adding the other points.
Stand-off Howarth had got the Scots up and running with two early penalty strikes for offside.
The visitors bounced back briefly to carve out a try through full-back Davies, which was goaled by Bowen, but Glasgow snatched full control to kill off the Welshmen before the interval.
Scrum-half Beveridge claimed the first try after powerful charges by Gareth Maclure and Scott Lawson, with Howarth slotting the conversion and adding a quick penalty.
Bowen clawed back three points, but stand-in skipper Logan burst into the contest, latching on to a long Howarth pass to score under the crossbar.
Howarth then jinked over for a brilliant solo effort that featured two sidesteps.
Logan burrowed through in the corner on the stroke of half-time, with Howarth keeping up his 100% conversion record.
He kicked his fourth penalty just after the restart as the Scarlets continued to toil.
But they refused to fly the white flag and sub Finau collected a long-range try, with Bowen adding the extras.
Scrum-half Phillips followed suit as Glasgow took their foot off the pedal - and the jitters set in when Murray was dismissed.
They were reduced further as Fergus Thomson was sin-binned - and Evans slid over to salvage a late consolation point for the Welsh.
Glasgow: C Shaw, R Kerr, A Craig, G Maclure, K Logan (capt), C Howarth,
G Beveridge, K Tkachuk, S Lawson, E Murray, A Hall, D Turner, C Mather, J Beattie, A Wilson.
Replacements: F Thomson, A Kelly, J van der Schyff, J Barclay, S Pinder, C Gregor, D Millard.
Scarlets: B Davies; G Evans, M Watkins, T Selley, A Havili; G Bowen, M Phillips; P John, M Rees, J Davies, V Cooper (capt), A Jones, J Mills, G Thomas, D Hodges.
Replacements: D Maddocks, A Gravelle, B Griffiths, A Powell, C McDonald, Aled Thomas, S Finau.
Referee: David Keane (Ireland)