However, the Scottish side will have to pull off minor miracles next week against champions Toulon if they are to be parachuted straight into the second-tier Amlin Challenge Cup, particularly after an error-strewn match which brought them just four points.
Chris Fusaro, the Glasgow captain, acknowledged his side's success in defeating an English team home and away in the same season but otherwise struggled to find positives after a turgid affair that offered up only three points in the second half. "It is a massive achievement, being the first Scottish team to do a double against an English side in Europe," said Fusaro.
"But we need to be grounded. Good effort, but the consistency wasn't there and we didn't quite click. We'll need to get it right for next week. We don't like making mistakes, but it is something that has crept into our game. Of course, we'll still throw the kitchen sink at Toulon. We are still hurting from that embarrassing game in France [a 51-28 defeat in October]."
There were a few more reasons to wince during an awful second half on Saturday. One such moment came in the dying seconds when free spirit Niko Matawalu, who had come on as a replacement for Henry Pyrgos, hoisted a kick into the afternoon sky rather than plonk the ball into touch as every team-mate was urging him gently to do. As it was 15-10, a try for Exeter would have at least earned the draw which sent both sides out of Europe altogether but, fortunately for the visitors, Glasgow turned the ball over on their own line to spare the Fijian scrum-half's blushes.
Exeter pressure would book-end the match as the contest started with a one-metre try under the posts - and under the Warriors forwards' noses - by Fetu'u Vainikolo not long after Henry Slade slotted a penalty. The Junior World Cup winning fly-half, who is now part of the England Saxons training squad, scored an early conversion too, but faded badly as the game wore on.
Indeed, Exeter struggled to create much else, perhaps a by-product of Rob Baxter, their head coach, trying to shake things up by replacing 10 players for this match.
Glasgow, on the other hand, replied instantly to the Exeter scores and finished the first half strongly. Sean Maitland scored in one corner - the opposite one from his own wing - after Sean Lamont collected a knock-on from the kick-off and started a passing move which ended with Ed Kalman popping a pass to Maitland like it was a medicine ball.
Lamont then scored his own try in the other corner and Stuart Hogg converted the score having taken over the duty from Jackson, who had run well but kicked poorly. Once Hogg slotted a penalty on the other side of half time and no-one else ever looked like scoring.
That was until Matuwalu's little misunderstanding. "It gave us a few minutes more tackle practice," added Fusaro. The line was delivered as a joke, even if his eyes suggested that he was a more than a little miffed.