Two penalties from Alex Davies helped a much-changed Welsh outfit reach half-time level at 6-6, and another penalty early in the second half briefly nudged the visitors in front.
But Grenoble, whose first half points had come from two Clement Darbo kicks, responded with a converted try by Vincent Campo, and Andrew Farley scored a second late try for the hosts to confirm victory.
The result leaves London Welsh in third place in Pool Five with just one win from three games. Grenoble top the group ahead of Stade Francais, who were due to play Cavalieri on Saturday.
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