Gregor Townsend's side may have left it late before a try from their replacement prop Jerry Yanuyanutawa sealed the result a sent them to the top of the table in combination with Munster's defeat to Benetton Treviso in Italy. However, it would have been a travesty if they had failed to claim the points against a visiting side who were rarely in the game after Glasgow wrestled control.
Not only was it the Warriors' first win over the former European champions since 2010, it was also their third successive victory, giving them their best start to the season since the Celtic competition went to a proper league structure a decade ago. Better still, Glasgow have still not conceded a try this season, and have now beaten both of last season's finalists in consecutive weeks.
However, Townsend, the head coach, was quick to play down talk of title credentials at the same time as paying tribute to the work of his players in defending for half-an-hour and then taking charge for the rest of the game. "That was down to effort," he said. "We did not panic when we let them back into our half, which was great to see. The forwards played really, really well but there are 19 games still to go. We have started well in terms of results and the performance in this game was much better than it has been but there is still a long way to go; what matters is the way you finish."
For the players, there was a sense of satisfaction. "It was a different game to last week - we created a lot more chances - but it is testament to us that we have not yet conceded a try," said fly-half Ruaridh Jackson. "We got stuck in and to get this win against a team that has lost once in the previous 20 matches makes us very proud. I wouldn't say it was revenge for last year's semi-final - we were pretty devastated the way that went and the way they played in getting into the final - but it feels good, really good.
"It shows we have carried on the momentum from last year but it does mean there will be even more pressure on us now because teams will be gunning to beat us. We have beaten the top two teams from last year, which his a great start, but we have to back it up."
What was really impressive was that Glasgow produced a victory last night despite not starting well in a cagey game where neither side was prepared to take a risk, knowing that one mistake could be all it took to decide it. The result was a desperately tight match that remained goalless until the sell-out crowd had been forced to endure 24 minutes of the teams feeling each other out in what was essentially a battle over territory.
Leinster seemed more able to control the ball, recycling endlessly but finding the Glasgow in no mood to buckle however patiently they kept the ball and whatever clever angles they tried to find as they tried to attempt the Scots wide and find space when they turned the ball back inside.
The Irish outfit did earn a penalty - strangely for a scrum offence against Ryan Grant, the Scotland prop, when it was clear Glasgow were winning the pushing contest - which Jimmy Gopperth, the former All Black brought in to replace Jonny Sexton, duly kicked.
Glasgow seemed to have blown their chances when Stuart Hogg was wide with two long-range penalty chances to equalise, but that only seemed to inspire them even more. They had a couple of half-chances, both lost to sloppy handling in the breakdown, before they eventually managed the breakthrough they had been hunting so desperately.
It started with the decision to reject a third tricky penalty shot at goal and go for the line out instead. A secure maul started them going before the ball shifted into midfield. Jackson got close to the line, a ruck formed and Chris Cusiter, the captain and inspirational scrum-half making his return to the starting XV a year after he damaged a shoulder, popped the ball up to Chris Fusaro, who had the space and strength to go over. Hogg converted and although he missed another long-range penalty with the last kick of the first half, Glasgow felt they had had the better of the first 40 minutes in what was never going to be a high-scoring game.
In the second half, Glasgow took total control in terms of territory and possession but Leinster have a defence capable of winning Heineken Cups and refused to buckle. Instead, they hauled themselves back within a point on a rare breakout when Gopperth kicked his second penalty.
However, they could not break the siege for long enough to seriously threaten again. Glasgow had further half-chances - Tommy Seymour was prevented from grounding the ball and both Jackson and Mark Bennett missed shots at goal that would have eased the pressure - until they again kicked for the touchline and mauled, and this time Yanuyanutawa piled round the side and straight over.
It was a narrow win, but all the more satisfying for that. Champions do not fall without a fight.