Leicester, comfortable 41-13 winners at home to Stade Francais, will have the advantage of playing the tie in England, at the City Ground, home of Nottingham Forest FC on April 24.
The all-French game in Paris was not decided until the final minute when Maxime Machanaud kicked his third penalty.
Juan Imhoff had scored Racing's try in the second minute, before one by Charles Ollivon six minutes later when the holders were 10-0 down.
Level by the interval, they led briefly 13-10 before being pegged back and then overhauled in the exciting finale.
Leicester went ahead in only the second minute of their game, with a try by Manu Tuilagi.
Two tries in three minutes just after the half-hour by Fiji international Vereniki Goneva and Freddie Burns then helped them towards a 24-6 lead at half-time.
The visitors, in poor form domestically, never looked like threatening Leicester's outstanding home record, which has now been extended to 13 wins in 14 games.
Former Leicester man Julian Dupay scored their only try in the second half but further scores by Goneva, Mike Fitzgerald and Telusa Veainu had the packed Welford Road crowd celebrating long before the end.
The line-up for the other semi-final was decided on Saturday when Saracens beat Northampton Saints 29-20 and Wasps had a dramatic last-kick victory over Exeter Chiefs 25-24.
The Saracens-Wasps tie will take place at the Madejski Stadium, Reading on April 23.
(Reporting by Steve Tongue, editing by Pritha Sarkar)
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