The Belgian seventh seed dominated the opening set in ferocious conditions but the match changed direction after 36-year-old Benneteau won the second set on a tiebreak.
Goffin was listless in the third set as Benneteau moved ahead and fell a break down in the fourth, saving a match point on his own serve at 3-5.
World number 59 Benneteau had an attack of the nerves as he tried to close out the match, throwing in a couple of double-faults to allow Goffin to level at 5-5.
But he got the job done in the tiebreak to reach the third round of a grand slam for the first time since 2013.
(Reporting by Martyn Herman; Editing by Amlan Chakraborty)
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