Midfielder Alessandrini poked the ball home on 88 minutes after keeper Yohann Thuram-Ulien had parried away a Vincent Pajot header into his path as the hosts booked a place in the last four against Montpellier.
St Etienne will take on Lille in the other semi-final.
Rennes started brightly, with Jonathan Pitroipa latching onto a Julien Feret cross to make it 1-0 from close range.
The 17-year-old Corentin Jean equalised for the visitors in the 22nd minute with a fine low shot from inside the box and Troyes produced a battling display until they cracked in the closing stages.
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- Jonathan Pitroipa