Trailing 19-8 with 11 minutes left, Irish threw everything at Leicester who were down to 14 men and scored their second try of the afternoon to reduce the deficit to four points.
But Leicester kept their cool, fought their way upfield and young fly-half George Ford sealed victory with his fifth penalty of the game, taking his tally to 17 points.
Leicester's victory keeps alive their hopes of retaining the cup after losing their opening game at Saracens. Their only try came from lock Rob Andrew, while Irish scored two through Anthony Watson and Tom Homer, with fly-half Ian Humphreys kicking a penalty and a conversion.
Prop prospect Simon McIntyre grabbed an opportunist try to help sink Worcester for the second time this season as Wasps opened their account in the LV= Cup with a 28-19 win.
The 21-year-old former England Under-18 player, who scored his first Wasps try in October's 10-6 win over Worcester in the Aviva Premiership, notched his second as they bounced back from last weekend's 22-17 away defeat by Cardiff Blues.
Winger Jack Wallace and replacement scrum-half Charlie Davies scored the other Wasps tries with third-choice fly-half Tommy Bell contributing 13 points with a five-from-five goalkicking display.
Worcester replied with tries from centre Alex Grove, right wing Andy Short and replacement left wing Ben Howard, with fly-half Joe Carlisle landing two conversions.
Wing Tonderai Chavhanga scored only the third-ever hat-trick of tries for Newport-Gwent Dragons but it was not enough to prevent his side's second defeat in the competition this season, 30-20 against Northampton.
Stand-off Ryan Lamb booted 20 points as the Saints turned on the style after the interval to overturn a 10-6 half-time deficit.
Flanker Phil Dowson and wing Jamie Elliott got the visitors' tries.
Northampton included England stars Lee Dickson at scrum-half and flanker Dowson for a clash they had to win after losing to Harlequins in the last play of the match a week ago.