It extends Leinster's unbeaten record in all competitions to 20 while Aironi will take solace from their performance against the league leaders, a week after going down by a point to a late Newport Gwent Dragons penalty try.
But it was not the cakewalk that was predicted. Leinster began brightly enough but Aironi, boosted by the return of Italy internationals Giulio Toniolatti, Roberto Quartaroli, Carlo Del Fava and Simone Favaro were imperious in defence throughout.
A fifth minute penalty from Madigan looked set to open the floodgates but two minutes later Aironi had a great chance to hit back, only for opposite number Naas Olivier to miss the target.
On 11 minutes, the warning signs for Leinster were apparent and when their constant pressure found no way through, had Tito Tebaldi gathered his attempted interception he had a clear path to the try-line.
Madigan then missed another effort from the tee before Tebaldi again almost crossed for the hosts as Aironi were showing little respect for their visitors' lofty reputation.
Leinster doubled their lead on 27 minutes however after a dangerous tackle from prop Lorenzo Romano and a minute later the returning Luke Fitzgerald wasted a glorious opportunity, knocking on after excellent work from Fionn Carr.
The visitors turned up the heat as the half-time whistle approached but further stoic Aironi defence limited Leinster to just one further Madigan penalty with the score 9-0 at the interval.
After the break Aironi soon got themselves onto the scoresheet, this time Olivier finding his range five minutes in and the Stadio Zaffanella crowd dared to believe when replacement Luciano Orquera landed his side's second penalty with 20 minutes to go.
But Leinster's class eventually told with Madigan making a superb break which brought with it a penalty, duly converted by the fly-half.
And as desperation kicked in for Aironi, Gilberto Pavan was send to the sin bin before Madigan hit penalty number five, finally dragging Leinster out of sight.
In the closing stages and with a nine-point cushion and a man advantage, Leinster smelled blood and finally scored the first try of the game on 75 minutes, replacement centre Reid intercepted in his own half and streaked clear.
Madigan added the extras to take his personal tally to 17 to ensure Leinster equalled rivals Munster's record of 14 unbeaten matches while the only sour note for Joe Schmidt's men was Jack MacGrath's yellow card in the final minute.
|Aironi Rugby Score Card|
|Leinster Rugby Score Card|