But, ahead of the return fixture next week, qualification for the quarter-finals now looks beyond Scarlets, who have lost all three games.
It was a shock that Glasgow ended up winning by so many, given that they took a while to get going and led just 10-3 at the break.
Duncan Weir converted his own try and added a penalty, while Steve Shingler kicked a penalty for Scarlets.
Another three-pointer from Shingler cut the deficit to four points but then it was all Glasgow.
Hooker James Malcolm got over on his starting debut from close-in and a spectacular hat-trick of tries for giant winger Taqele Naiyaravoro followed . Replacement lock Tim Swinson closed the match with the sixth Glasgow try.
Weir converted the first of these scores and his replacement Finn Russell the next pair and the final one.
The Scotstoun pitch that had been unplayable seven days earlier looked very soft in certain spots as Glasgow kicked off on a fine still mid-winter afternoon.
Glasgow got off to a flying start with a try in just the sixth minute from stand-off Weir that he converted himself.
Scarlets, who had their fair share of first-half possession, replied five minutes later with a penalty from Shingler.
In the middle period of the half Glasgow passed up three good chances to forge ahead.
The first was the clearest when, following a clever kick from centre Alex Dunbar that put the visitors' defence in a daze, full-back Stuart Hogg failed to hold a simple interception with the line seven metres away.
Then Naiyaravoro caught a Scarlets clearing kick and squeezed in, but was judged to have gone out of play in the process.
Finally, a charge down by Adam Ashe of a clearing kick bounced cruelly for the home side behind the Scarlets' line.
In the last minute of the half Glasgow scrum-half Mike Blair's quick tap led to a penalty that Weir struck home to give Glasgow a 10-3 half time lead.
Scarlets made a smart restart and a move started with a break by winger Mike Tagicakibau ended with a simple penalty for Shingler.
But Scarlets remained haunted by conceding penalties. Weir put one to the corner in 51 minutes and the line-out drive ended with a try for hooker Malcolm on his starting debut, which Weir converted.
Five minutes later the outcome was decided. From a standing start Naiyaravoro knocked two Scarlets aside as he finally got his try, converted by substitute Russell.
Scarlets still had the spirit to have replacement Jack Condy held up over the Glasgow line.
In the 70th minute, however, Naiyaravoro was all on his own on the right wing to collect, and run in for the bonus point, a cross kick from Russell.
Four minutes later he secured his hat-trick from an interception that went unconverted, and then r eplacement lock Swinson closed the match with the sixth Glasgow try, converted by Russell.