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The hosts got off to a lightening fast start, scoring three tries inside the opening 15 minutes, including two in the space of two minutes for Miles Benjamin.
However, the French side got themselves back into contention before the break as Mamuka Gorgodze and Timoci Nagusa capitalised from a strong spell of possession to both add tries, leaving the visitors just six points back at half time.
Pierre Berard gave the visitors some late hope, grabbing a brace of tries inside the final 15 minutes, but they could not prevent the Aviva Premiership champions from securing the win.
The result leaves Leicester in second place in pool five, while Montpellier remain third.