They took the lead on two occasions in an error-strewn opening half, but came away with just a losing bonus point in the end.
A well-executed move created a try for Glasgow full-back Peter Murchie before the break, and impressive prop Ryan Grant added a second touchdown coming up to the hour mark.
The Scots held on to take the points in a tense finish, with New Zealander James Marshall's converted try and penalty reducing the margin to just a single point.
Speaking after his side's first away win of the season, relieved Glasgow head coach Sean Lineen said: "It's just great to get that winning feeling away from home.
"It's always hard no matter who you're playing. Both teams were desperate but I'll take the four points."
In humid conditions, Aironi signalled their intent with an early barrage of attacks. From one such assault, winger Giulio Toniolatti's neat chip over the top should have resulted in a try, but centre Matteo Pratichetti knocked on with the line at his mercy.
Scrum half Tito Tebaldi sent over a ninth minute penalty for the opening score. The gloss was taken off that by the sin-binning of flanker Josh Sole for a foot trip on Murchie, just two minutes later.
However, Glasgow failed to score while the Italians were down to 14 men. Out-half Ruaridh Jackson missed a penalty from just inside the Aironi half and Tebaldi also failed with his second attempt.
Although Aironi were retaining possession well, handling errors continued to be a cause of concern. Another knock on spoilt a high intensity attack, and Jackson got Glasgow back on terms in the 28th minute.
Warriors centre Federico Aramburu had the ball ripped away from him just as he looked set to break the try deadlock. Play was brought back for an earlier infringement, but Jackson's shot at goal was wayward.
Tebaldi landed his second penalty to restore Aironi's lead at 6-3. Murchie's came at a crucial stage for Glasgow and gave them added confidence for what was an attritional second half.
The Warriors moved the ball out to the left at pace and with the Aironi defence stretched, Aramburu's delightful pass unleashed Murchie for a run to the line. Jackson's conversion made it 10-6 for half-time.
Lineen made four changes for the second half, bringing Hefin O'Hare, Colin Gregor, Dougie Hall and Johnnie Beattie into the fray.
Aironi boss Rowland Phillips also turned to his bench, but they fell further behind when Grant, who carried well at close quarters, was driven over for a 56th minute.
Jackson added the extras and Glasgow were on the right road to that precious away victory, leading 17-6 with just over 20 minutes remaining.
They were rocked back on their heels though, only three minutes later. Aironi worked an overlap for themselves and Marshall backed himself to score and did just that. The out-half converted his own try.
Marshall followed up with a 64th minute penalty. Aironi's powerful pack, led superbly by Marco Bortolami, Nicola Cattina and Nick Williams, kept hammering away at the Glasgow defence, desperately trying to find a chink of any kind.
Glasgow were unyielding and maintained their lead, while young replacement Duncan Weir missed a late penalty attempt.
Aironi's brave bid for a second successive triumph came to an end when a terrific tackle from replacement Jamies Eddie knocked the ball free and into touch for the final time.
|Aironi Rugby Score Card|
|Glasgow Warriors Score Card|