The Saints, tournament runners-up to Leinster in 2010, trailed 15-0 before they turned the Pool Four clash on its head through three tries in six minutes either side of half-time.
Centre George Pisi touched down twice, while fly-half Stephen Myler's try just after the break nudged Northampton ahead for the first time before wing Vasily Artemyev's late effort sealed it.
Myler and Ryan Lamb each booted a conversion to help wipe out an alarming deficit as Saints conceded tries to Glasgow flanker Josh Strauss and wing Sean Lamont after centre Peter Horne kicked an early penalty and converted Strauss' try.
Myler, though, missed four shots at goal that could easily have proved costly, and there is a lot for Saints to work on before they tackle Castres in France next Friday night.
Glasgow did not even gain the consolation of a losing bonus point, but they will feel that should have been the minimum return from their journey south.
Their forwards' appetite for physical contact - epitomised by Strauss and skipper Al Kellock - at times rocked Saints on their heels, and it took all the Aviva Premiership leaders' poise to retrieve the situation.
But rugby director Jim Mallinder will know that long spells of Saints' performance were below-par, with the Castres trip likely to be a key fixture in their European campaign.
England hooker Dylan Hartley returned to skipper Northampton after recovering from an eye injury as Saints targeted making an immediate statement of their European ambitions two years after reaching the final.
England coaching trio Stuart Lancaster, Andy Farrell and Graham Rowntree were among the onlookers, together with Scotland boss Andy Robinson and his assistant Scott Johnson, as both countries continued formulating plans ahead of next month's autumn Tests.
Glasgow, eager to show Scottish rugby in a far more promising light than Edinburgh following their 45-0 home defeat against Saracens yesterday, dominated early territory and possession.
But after hardly seeing the ball for 10 minutes, Northampton almost struck an opening scoring blow when wing Ken Pisi sprinted clear from inside his own 22. Glasgow, though, had enough defensive numbers to thwart the danger.
Ken Pisi was involved with Saints' next scoring opportunity after Glasgow went in front through a Horne penalty, but this time he could not ground the ball under pressure from a superb Lamont tackle.
And Glasgow stormed back upfield, launching a succession of drives deep inside Saints' 22 that ended with Strauss touching down. Horne converted, and Northampton had it all to do, 10-0 adrift.
Northampton were visibly frustrated at their failure to finish chances, and matters deteriorated eight minutes later when more impressive work by Glasgow's forwards and some slick passing from their backs ended with ex-Saints player Lamont powering over.
Horne could not add the touchline conversion, but Glasgow's 15-point advantage stunned the home crowd during a half the visitors looked to end by pushing for another try.
But Northampton dug deep when they had to, stunning Glasgow with two tries during first-half injury time.
George Pisi's opener came after his slashing midfield break sparked a period of concerted attacking pressure, then he finished off superb approach work from his brother Ken as Glasgow's defence found itself scattered to all parts.
Myler converted Pisi's second try, and Saints trooped off just three points adrift when it could have been so much worse.
Glasgow were forced into making an interval change when fly-half Ruaridh Jackson, who suffered a knock during the opening 40 minutes, was replaced by Niko Matawalu, with Horne switching to number 10 duties.
But they were powerless to halt increasing Northampton momentum, which was emphasised by a 45th-minute charge from prop Soane Tonga'uiha that created a platform for Myler to score.
It was an impressive revival by Saints as Glasgow, for the first time in an entertaining contest, were forced to find answers to some serious questions.
And when Artemyev sprinted over six minutes from the end of normal time, there was no way back for the visitors, although they could yet have a considerable say in the outcome of a group that also includes Castres and Ulster.
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