With Clermont Auvergne and Ulster making up the remainder of Pool Four, the two-time European champions would have targeted a maximum-points haul in northern Italy.
But despite dominating the first half they failed to grab the necessary four tries.
Tom Croft and Alesana Tuilagi crossed for tries in the rampant first half but Leicester took their foot off the gas following the interval and it was not until the 80th minute that Niall Morris crossed for their third try.
It is 10 seasons since Leicester claimed back-to-back Heineken Cup titles and they enter their latest assault on Europe following a dismal start to their domestic campaign.
Hit by an excess of injuries and international call-ups during the recent World Cup the Tigers won just two of their opening eight Aviva Premiership fixtures.
They threw away a 17-point led against London Irish last week as the Exiles stormed back in the final 20 minutes to salvage a draw.
England fly-half Toby Flood admitted the Tigers let head coach Richard Cockerill down and called on his team-mates to make amends today.
Cockerill made three changes to his side with Horacio Agulla the latest to join an extensive injury list which also includes Ben Youngs, Jordan Crane, Anthony Allen, Manu Tuilagi and Billy Twelvetrees.
The Tigers were more than aware of the threat Aironi pose after they caused the biggest upset of last year's competition with a slender 28-27 victory over Biarritz.
Leicester got off to a flying start at Stadio Brianteo in Monza and surged into a 13-0 lead within 15 minutes thanks to a pair of quick-fire penalties from the boot of Flood and a try from Croft.
With Sinoti Sinoti in the sin-bin Leicester stretched the hosts either way and after working the overlap Alesana Tuilagi released Croft to squeeze in for a converted try.
Cockerill's men continued to threaten in attack and they clinched a second try after 25 minutes when Tuilagi took a short ball off Flood and crashed over.
The England fly-half dissected the posts to take the score to 20-0 but Aironi quickly hit back with a penalty.
They continued their resurgence with a second Orquera penalty on 33 minutes to bring the scores to 20-6 at the break.
Flood struck the uprights within two minutes of the restart but he made no mistake three minutes later to stretch Leicester's lead to 23-6.
Orquera quickly struck back with his third penalty and following a sustained period of pressure he slotted another on 68 minutes to make it 23-12.
That appeared to stir the Tigers to life and they should have grabbed their third try on 72 minutes but unmarked Tuilagi failed to hold onto Matt Smith's wild pass.
They continued to hammer away at the hosts but could not break Aironi's unforgiving defence until it was too late.
Following some strong carrying by Thomas Waldrom, the visitors worked the overlap and Leicester captain Geordan Murphy threw a pinpoint miss pass to fellow Irishman Morris for a touchdown in the corner.
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