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Shane Geraghty scored two penalties early in the game to hand his side a six-point advantage in a tight contest.
Just before the break, the visitors made a crucial breakthrough as prop Halani Aulika powered over the line for the first try of the match, but two kicks from Freddie Burns kept Leicester in the game.
Burns and Geraghty exchanged further scores after the break to keep the contest close, but the home side managed to drag themselves within striking distance when centre Niki Goneva went over the tryline.
The boot of Geraghty put his side further ahead when they were awarded a penalty in the 61st minute, but Owen Williams brought his side level in the 73rd to set up a tense finale.
However, Irish won a penalty in the last minute of the match and Geraghty managed to hold his nerve to fire his team to victory.