Instead, they engaged reverse gear.
They had gone into Saturday's match already certain to finish bottom of their pool and those who know what they have been working towards in recent years had the courage to recognise there was little consolation in this win.
"If you take that game in isolation and the fact that [Northampton] were coming up here with a chance of going through, it's a brilliant win, but we hadn't won any of our previous games and we should never have been in that position," said Al Kellock, who has captained the side throughout his time with them.
"We've done this before, winning one of these last two games and it's good to win them, but not having any wins before this is not acceptable and we said that before the game. We've got character, always have and I believe now more than any time that we've got talent. We've got the ability."
In saying so, Kellock tacitly acknowledged that while previous funding restricted Glasgow to aim for an over-achievement in avoiding the bottom spot in each pool, taking that place this time around with their new-found resources accentuates their under-achievement.
"We can't just save it for the last game," Kellock went on. "We've got to make sure that we go into these last two games pushing for the quarter-finals. I said to the boys that it's a long time to wait for the Heineken Cup again, we've got to remember that losing is not acceptable.
"When losing becomes unacceptable there's a change. You get that little extra edge. We've got it in the league where we go into every game knowing we should win and expect to win. I don't know if we're missing that slightly in Europe. You look at our group and it's very good, but we're a good team. However we've got to prove that consistently and not just in one-off matches."
That, though, was all that was left to them on Saturday and Kellock acknowledged the importance of the performance when placed in an international context – many of the players will be facing eachother at Twickenham when the RBS 6 Nations begins in a fortnight.
"It was massively important," he said when asked how it related to the Calcutta Cup opener. "I said at half-time that men go out and win that game. It's easy to play our role, just go out for 40 minutes and push Northampton hard when Northampton, if they win it, qualify and Glasgow just bow down, but men go out and win that game and we did that."
It did not look like they would as Northampton dominated possession in the opening half hour but Kellock felt his men had responded well to the insulting way the English side repeatedly chased tries rather than kicking the penalties that would have put them in control.
"It annoys me when a team comes up here and boots the ball in the corner... when their 10 gets the ball and looks at the corner instead of the posts that gets under my skin." He felt they could still have defended better than they did in conceding three tries on the day, but there was plenty evidence of why Scotland's interim head coach Scott Johnson has emphasised the importance of Glasgow's defence in his plans since the concession of a single try, scored by Dom Waldouck on the half hour, was meagre reward for Northampton's first half dominance.
It had been the only score of the half but Glasgow upped their effort after their captain's interval remonstrations to build on two early penalties Ruaridh Jackson's two early penalties had been countered by a solitary strike from Steve Myler.
Josh Strauss's touchdown early in the second half gave Glasgow the lead though Northampton did regain it with George Pisi's try midway through the half but the irrepressible Niko Matawalu, rightly adjudged man of the match for his all-action contribution at scrum-half and latterly on the wing, put Glasgow back in front before the visitors belatedly realised that the win was no foregone conclusion.
Ben Foden's 71st-minute try produced a level scoreline at 20-20, but Horne's brilliant swerving, side-stepping opponent fending run was a spectacular way to finish and a very opportune time for one of the nine uncapped Glasgow players named in the Scotland squad last week, to catch the eye.