Saints produced a stunning team display as they claimed a memorable 26-15 win at Scotstoun Stadium on Saturday night.
The valiant victory had a distinctly Samoan flavour as Ahsee Tuala scored twice and George Pisi also dotted down.
Stephen Myler added 11 points with the boot in a fine kicking display as Jim Mallinder’s men made it two wins from as many games in this season’s Champions Cup.
Glasgow had won their last six games when hosting an English side in Europe’s premier club rugby competition, but they had no answer to Saints’ power and panache.
The Warriors did respond through tries from Peter Horne and Josh Strauss, but Saints deservedly travelled home with four precious points in their collective suitcase.
Glasgow had gone into the game as favourites, with last season’s Guinness Pro12 champions formidable on home soil.
They were also refreshed having not played for two weeks, following the postponement of their game at Racing 92 a week earlier.
And they started with three points from fly-half Finn Russell, who atoned for missing his first penalty attempt.
But Saints issued a rapid riposte as Glasgow failed to deal with a high ball and centre Pisi stretched out an arm to score.
The try was greeted with a roar from the sizeable Saints following and they were even happier when Myler added the extras.
Saints were showing belief and Glasgow were making a couple of errors, inviting the English side into their territory.
And after a successful shove at a scrum five metres out, Myler hit the mark with his first penalty of the evening.
Glasgow were struggling in the scrum and Saints were turning up the heat, forcing Ryan Grant to infringe, with the loosehead prop heading for 10 minutes in the sin bin.
Myler was soon adding another penalty to the tally as the home fans’ frosty relationship with referee Pascal Gauzere continued.
They did turn the tide briefly, with Saints infringing on a couple of occasions, but the natives were to grow even more restless on the half hour mark.
Scotland full-back Stuart Hogg missed a tackle out wide, leaving Tuala free to glide in for the score.
Myler missed the conversion, but it was still a stunning start for Saints and a terrible one for Glasgow, who welcomed Grant back on to the field at 18-3 down.
But Saints were soon to pay for a moment of sloppiness of their own as Mike Haywood threw the ball dead, giving Glasgow a scrum.
They used it to score as Horne, who scored the winning try when Saints last visited Scotstoun, in January 2013, latched on to a pass and dotted down, to the delight of the home fans.
Russell converted, but Saints were to have the final say of the half as the forwards won the scrum against the head and Gauzere gave another penalty, which Myler slotted.
Saints were forced to do some defending at the start of the second period as Glasgow twice went for the corner, but defend they did, with great defiance.
And after Kahn Fotuali’i, on at half-time for Lee Dickson, pushed Glasgow back with a fine kick, Ben Foden picked up the home side’s clearance and launched a superb break.
It ended with the full-back picking out Tuala with a sublime grubber kick, and Saints scored again.
Myler hit the post with the conversion and Saints were soon under the cosh again, with the home fans baying for blood.
They got it, as Luther Burrell was sin-binned, and from the next attack Strauss powered over for a try.
Russell missed the conversion to leave the gap at 11 points with still more than 20 minutes to go.
Considering they had conceded at the start of Burrell’s sin-bin period, Saints had done well to limit the damage to five points when he returned to the field.
And Glasgow simply could not get through in the final 20 minutes as Mallinder’s men held out superbly, even trying to push for a bonus-point fourth try towards the end.
As it was, they made do with a fantastic four points from their trip to Scotland.
Glasgow: Hogg (Naiyaravoro 53); Seymour, Bennett, Horne (Vernon 63), Lamont; Russell, Blair (Hart 73); Grant (Reid 53), MacArthur (Mamukashvili 8), Puafisi (Fagerson 58); Nakarawa (Swinson 68), Gray (c); Harley, Favaro (Wilson 53), Strauss.
Saints: Foden; Tuala (Elliott 62), G Pisi, Burrell, North; Myler, Dickson (c) (Fotuali’i 40); A Waller (64), Haywood, Brookes (Hill 64); Paterson (Harrison 68), Day; Gibson (Fisher 72), Wood, Dickinson.
Replacements not used: Williams, Hanrahan.
Referee: Pascal Gauzere (France)