The 22-year-old was instrumental in Leinster's best attacking play, scoring the first of their four tries in the 48th minute.
The province had been in a sticky situation after Nick Williams' converted try earned a surprise half-time lead for the Italians, who have yet to win an away game in the RaboDirect PRO12.
But Leinster knuckled down in the second half and a last-minute try from stand-in captain Rhys Ruddock was the reward for the persistence shown in difficult wet conditions.
Despite having the better of the early exchanges, Aironi faltered in the hosts' 22 as three knock-ons in the opening 10 minutes prevented the visitors from getting the first points.
Leinster got their chance 16 minutes in when Aironi were punished as the home forwards mauled through with intent. However, Isa Nacewa failed to capitalise with the full-back kicking his penalty wide of the right hand post.
Leinster's backs failed to make the most of their opportunities with ball in hand. David Kearney was penalised when taking the ball into contact, Fionn Carr failed to gather a Nacewa offload with the try-line metres away and then when Luke Fitzgerald darted inside the Treviso defence, he wrongly decided to take the ball into contact with Kearney available on the overlap.
In the 23rd minute Aironi were reduced to 14 men when right winger Sinoti Sinoti was sin-binned. Nacewa pulled the resulting penalty wide of the left upright and minutes later reproduced an identical miss when a Cillian Willis charge-down led to another penalty opportunity.
Aironi responded with a period of sustained pressure that brought about the game's first points. With five minutes left before the break, the Italian pack used lineout ball to drive towards the Leinster line and Williams was able to touch down. Tito Tebaldi converted to give the visitors a 7-0 lead.
Leinster responded immediately with Rhys Ruddock making the initial break and Schmidt's men inched closer to the Aironi whitewash. It was as close as they were going to come however, as their push to the line was held up short and Aironi cleared their lines.
Madigan, who came on for debutant Noel Reid, made an immediate impact after the resumption. He first found Kearney on the wing in a phase which resulted in a penalty for the home side.
With Nacewa having missed three penalty attempts, Madigan was asked to go for the corner and when the ball finally came back it was Madigan himself who split the Aironi defence before touching down.
Nacewa's conversion levelled the tie and after a brief spell of Aironi pressure, it was Madigan again who broke the first tackle and set the Leinster attack in motion. The lively replacement then linked with the supporting Nacewa to finish the move in the 58th minute, cutting in from the wing to score to the right of the posts.
The try went unconverted and it took Leinster another 10 minutes to cement their victory when Fitzgerald scored a breakaway try. The Ireland international collected a stray Gilberto Pavan pass on halfway and dashed in under the posts. Nacewa added the extras for a 19-7 advantage.
With nine minutes remaining, Carr cut inside from the wing before a backhanded offload found Fitzgerald. Leinster recycled the ball and although Madigan's looping pass to wing found Kearney, a subsequent knock on spoilt the promising move.
With time almost up, Ruddock pounced for his try following a five-metre scrum. Aironi had taken the ball behind their own goal line and were forced to touch down, providing Leinster with the perfect attacking platform to gain the bonus point.
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