They had nothing but pride to play for, but even that energised their performance as they crushed Northampton's (admittedly slim) hopes of European progress.
Horne might dine out for years on the swerving run that took him past half the Northampton team and ended with the replacement centre diving over beneath the posts. It was an act of rugby genius, but it was only possible after the act of rank stupidity that Saints prop Alex Waller had committed a few seconds earlier.
With Northampton pressing for the fourth try that would bring them a bonus point, Waller threw a punch at a Glasgow player. He was about five yards from the Glasgow line, and about six inches from the touch judge's nose. The official alerted the referee, the referee reached for the yellow card, and Northampton's European dream was over.
Horne, however, still wanted to add salt to their wounds. The clock had gone past the 80-minute mark, and the sensible thing would have been to boot the ball into the stand, but Horne, newly inducted into the Scotland squad, took it for an adventure instead. He dummied a pass one way, stepped past a player the other way, then shimmied and swerved past the flailing defence on his glorious run to the line. If Glasgow coach Gregor Townsend, a former Northampton player, felt a pang of sympathy for his old club then he managed to disguise it rather well.
"This is a reminder of where we should be, winning games in the Heineken Cup," he said. "That's what we're here to do. I think our forwards were outstanding, up against one of the best packs in Europe. In the second half we had most of the ball and were moving it around well. It was great to see that passion and effort in the second half.
"I think we have shown it at times in every game. We have shown it in patches. We just haven't had the 80 minutes and the winning performance. I'd like to focus on the positive feelings rather than frustration, but ultimately it has not been the season we wanted."
For the first 40 minutes, the best of Glasgow was seen in defence. It was last-ditch stuff as well, but it was impossible not to warm to their determination, particularly when Stuart Hogg pulled off a magnificent try-saving tackle on Saints winger Jamie Elliott after 17 minutes. Warriors' line broke just once, when Dom Waldouck scythed through for the first try just after the half-hour mark.
After the turn, though, Glasgow began to show what they could do going forward. They had butchered a glorious chance shortly before the break when Sean Maitland tripped just a few yards from the undefended line, but Josh Strauss made amends in the 46th minute when he barrelled over from a lineout move after Al Kellock had made the initial dent.
Northampton struck back in what was becoming a see-saw contest when George Pisi slid over 12 minutes later, but Glasgow responded again when Niko Matawalu, who was superb throughout, ripped the ball away from Ben Foden near halfway, planted his foot on the gas and raced off for a score.
Again, Northampton fought back, Foden going over in the 72nd minute after his side stole Glasgow's scrum ball. With eight minutes left, and with Saints desperate for one more touchdown, a dramatic finale looked likely. But few would have guessed just how dramatic it would be.
Glasgow: S Hogg, T Seymour (S Kennedy 57), B McGuigan (P Horne 55), G Morrison, S Maitland; R Jackson (S Wight 35), N Matawalu; G Reid (R Grant 61), P MacArthur (D Hall 51), M Low, T Swinson, A Kellock (captain; T Ryder temp 26 - 29), J Strauss, J Eddie, R Wilson (T Ryder 46)
Northampton: B Foden, J Wilson (T May 56), G Pisi, D Waldouck, J Elliot; S Myler (R Lamb 56), L Dickson; S Tonga'uiha (A Waller 56), D Hartley (M Haywood 68), P Doran-Jones (T Mercey 55), S Manoa, C Lawes (R Oakley 51), C Clark (M Sorenson 37), T Wood, P Dowson.
Referee: L Hughes (Wales).