Warriors, who are now unbeaten in five RaboDirect PRO12 games, added further tries after the interval through Chris Fusaro and Murchie, but left it too late to seal what would have been a valuable five points, particularly given Ospreys' surprise 23-22 win over league leaders Leinster.
Aironi, who remain rooted to the foot of the RaboDirect PRO12 table, offered little in terms of attacking threat and two first-half penalties from Luciano Orquera was all they could muster.
It was a bright start by the Warriors, and it took just three minutes for Duncan Weir to open the scoring with a nicely-taken penalty from just outside the 22.
Ryan Grant almost forced his way over in a relentless spell of pressure from the hosts that resulted in another penalty - slotted home by Weir - for a 6-0 lead.
The resolute Aironi defence continued to frustrate Lineen's side on their own line - restricting Weir to another penalty despite total territorial dominance throughout the opening 25 minutes.
In one of their first forays into the Warriors half the Italians grabbed three points of their own through the boot of Orquera before the Italian fly-half again sailed an effort between the posts on the half hour mark.
With tension building at Firhill a superb Seymour break through the Aironi line five minutes before the interval gave the Scottish side their opening try - under the posts - which Weir converted for a 16-6 half-time advantage.
Four minutes after the restart Weir was again on the scoresheet with a 30-metre penalty but that elusive second try evaded them until Fusaro burrowed his way over on the hour mark after a powerful maul.
Fusaro was adjudged to have crossed the whitewash by the TMO but Weir failed to add the extras with his first wayward kick of the evening.
The flanker was then denied his second try of the season with 15 minutes remaining as Lineen's side hunted down the bonus point.
And with eight minutes remaining it was within touching distance after a lightning attack from the breakdown by Samoan winger David Lemi opened up space on the left for Murchie to easily cross.
Scott Wight's missed conversion mattered little before Lemi and Troy Nathan both came close as the Warriors desperately tried to steal a valuable fourth try but, just as they did in the early stages of the first half, Aironi held firm.
|Glasgow Warriors Score Card|
|Aironi Rugby Score Card|