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The 18-year-old Fils, half the age of his opponent, won the first-round match against his fellow Frenchman 7-5 7-5 in Montpellier.
Fils will face another battle between youth and experience when he takes on Spaniard Roberto Bautista Agut in round two.
The 163-ranked Fils served eight aces and broke Gasquet four times to move into the next round
Nikoloz Basilashvili will face either defending champion Alexander Bublik or Gregoire Barrere in the last 16 after beating Constant Lestienne 6-3 (7-3) 7-5 6-2 in Monday's other match.