That would mean they couldn't finish as the bottom Welsh side in the Celtic League, regardless of what the Dragons and Ospreys do in their remaining games, and so they would ensure avoiding an unwanted play-off against Italian opposition in June.
Wales star Alix Popham, whose two tries paved the way for Tuesday's 18-6 victory over Heineken finalists Munster is crystal clear about the importance of Euro qualification.
"Being in the Heineken Cup next season is essential," said the No.8. "It's the pinnacle for the Scarlets. We've done well in it in the past and want to be in it again.
"We're coming to the end of the season and the boys are feeling tired, but we've got to lift ourselves for the remaining challenges.
"It will be tough against Glasgow, with it being our third game in six days, and they are a hard side to play away from home but, hopefully, we can get the points."
Forwards coach Paul Moriarty is determined to wrap up a Heineken spot tonight rather than wait for next weekend's final game at home to Ulster, which will be the Stradey farewell for Wales-bound coaches Gareth Jenkins and Nigel Davies.
"Qualifying for Europe is in our hands and the sooner we put it to bed the better," he said. "For a proud region like the Scarlets, with its supporters and it's traditions, it's vital we are in the Heineken Cup.
"If we go to Glasgow and win we can relax a little bit before Ulster come to Stradey for Gareth and Nigel's send-off. Obviously, the two of them have put a lot of time and commitment in with the Scarlets and they deserve a big farewell party. If we don't qualify this weekend, it will drag on longer and we don't want that."
The Scarlets show just one change from the side that beat Munster, with Wales wing Mark Jones returning from injury in place of Tal Selley, while rookie Craig Evans continues at fly-half. One-from-bottom Glasgow, who have lost their last six league games, must win and hope Borders lose in Ulster if they are to retain their own slim hopes of Heineken Cup rugby.
Glasgow: C Shaw; S Davey, H O'Hare, A Henderson, R Lamont; D Parks, C Gregor; K Tkachuk, F Thomson, B Prescott, J Eddie, D Turner, J Petrie, J Beattie, D Macfadyen (capt). Replacements: S Lawson, S Corsar, C Hamilton, J Barclay, I Monaghan, G Morrison, G Staniforth
Scarlets: L Byrne; G Evans, R King, G Evans, M Jones; C Evans, C Stuart-Smith; I Thomas, M Rees, C Dunlea, V Cooper (capt), H Louw, I Afeaki, A Popham, D Jones. Replacements: J Davies, A Gravelle, A Jones, P John, J Mills, D Peel, P Mackey
Referee: George Clancy (Ireland)