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.
Irish, who started their cup campaign with a win over Sale, started brightly and led 8-0 after nine minutes thanks to a penalty from fly-half Humphreys and a fabulous try by full-back Watson. It was made by winger Topsy Ojo, who capped a brilliant break down the right wing with a well-timed inside pass for Watson to score. Humphreys failed with the conversion.
But Leicester hit back and dominated the rest of the half. They led 10-8 after 24 minutes, a lead that could have been bigger but for two missed penalties in the 16th and 30th minutes from Ford.
Leicester scored 10 points inside four minutes, their try coming after some fierce pressure on the Irish line. Humphreys' charged down kick led to a line-out five metres out, the Tigers pinched it and Andrew surged infield before barging his way over near the posts for his first try for the club on his fifth appearance to give Ford an easy conversion.
Ford then slotted a 40-metre penalty after locks George Skivington and Matt Garvey were warned for taking out Brett Deacon in the air at a line-out.
And with a minute to go the 19-year-old Ford, who was the International Rugby Board's Young Player of the Year in 2011, capped a fine first half with a 40-metre penalty to stretch Leicester's lead to 13-8.
Leicester piled into the Irish at the start of the second half and got their reward for some strong attacks when the referee awarded them a penalty in front of the posts which Ford kicked in the 49th minute to make it 16-8.
Irish's best moment came when Humphreys made a sizzling counter-attack from his own 22 but his pass to scrum-half Darren Allinson, who looked likely to score, went forward.
Leicester, with young prop Fraser Balmain having a fine game, also dominated the scrums and they earned themselves a penalty 56 metres out when they demolished the Irish pack and Ford banged it over.
Allinson put some much-needed fizz into Irish with a fabulous break from a ruck before feeding Garvey, who looked certain to score before Micky Young brought him down 10 metres out.
Irish were eventually awarded a penalty near the posts but somehow Humphreys contrived to miss it.
Leicester had prop Boris Stankovich sent off for a high tackle, much to the annoyance of the 17,569 crowd, and the extra man told with 11 minutes left when replacement Homer squeezed over in the left corner. Humphreys' brilliant conversion reduced the arrears to four points.
Ford, however, sealed an important win for Leicester with his fifth penalty with two minutes left.
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