Aironi's contributions to the scoreboard came from the boot of Naas Olivier who kicked six first-half points but the Warriors - led by Weir - ultimately bagged themselves a comfortable win.
The Warriors drew first blood early on when Aironi were penalised at the ruck and Weir kicked the penalty but the Italians were soon level after Troy Nathan was caught offside.
Weir's opposite number Olivier made no mistake with the easiest of penalties on five minutes.
But his next points were far from comfortable. After a period of sustained pressure from the home side the South African fly-half notched three points from a tough angle after the Warriors were caught offside after 11 minutes.
The kicking war continued with Weir taking the lead, first levelling things with another penalty before making it 9-6 on 22 minutes - a three-pointer again doing the damage.
After his dead-eye start with the boot Weir showed his human side with the half-hour mark approaching - missing a drop goal before coming up short with a lengthy penalty attempt.
But the 20-year-old fly-half came roaring back from those disappointments and made sure his, and Glasgow's, first-half ended on a high by notching a drop goal with the last kick of the half to make it 12-6.
The Italian side were seemingly handed the perfect opportunity to get back into the game early in the second half when Johnnie Beattie's troubles continued when he was yellow carded for coming over the top of a ruck.
The Scotland international was overlooked for the World Cup squad earlier this year and having been dropped last weekend was looking for redemption in Italy but it failed to materialise.
Beattie took some solace from the fact that Olivier missed the resulting 48th-minute penalty and another three-point effort seven minutes later.
The Warriors stretched their lead to nine points on 62 minutes when Weir took his personal tally for the afternoon to 15 with another three-pointer after Aironi were caught offside.
And he was at it again ten minutes from time when the Italians were penalised for bringing down the scrum - this time slotting impressively from distance to make it 18-6.
Despite Weir missing a late penalty the Warriors easily saw out the final stages to take the win - their fifth of the season.
|Aironi Rugby Score Card|
|Glasgow Warriors Score Card|