The victory means that the Tigers will now face last year's champions Harlequins in the semi-finals.
Toby Flood levelled the game up shortly after with a well-hit penalty.
The Tigers then turned the game around when Tom Croft went over in the corner to give them an 8-3 lead.
Leicester scored another try three minutes later, when Ben Youngs found a gap before evading challenges to go over to give the hosts a 13-3 half-time lead.
Leicester added a third try 10 minutes into the second half, when Matthew Tait went over from a Manu Tuilagi pass, allowing Flood to convert to give the Tigers a 20-3 lead.
It didn't take long for Irish to respond, as four minutes later some great play by the visitors' pack allowed Alex Corbisiero to pick up and drive himself over the line for 20-10.
Vereniki Goneva allowed Tigers some more breathing space when his try extended the home side's lead to 27-10, before Flood scored to give Leicester a 22-point advantage.
With eight minutes left to play, Irish found their second try of the match as a perfectly weighted chip from Shane Geraghty was gathered by Topsy Ojo to score in the corner to bring the score to 32-15.
Marland Yarde managed a late try for the away team, but Tigers picked up a 32-20 win.