There was little to separate the players in the first set on Court Suzanne Lenglen, with Gill breaking back immediately after dropping serve in the third game.
But Jacquet opened up a 3-0 lead in the tie-break and Englishman Gill could not claw his way back as the Frenchman secured the opener to the delight of the home crowd.
Jacquet carried his momentum into the second set and struck the decisive blow at 3-2 before serving out victory on his second match point four games later.
Of the six British players in both singles draws, only Norrie is seeded.
Norrie will open against Paraguay’s Adolfo Daniel Vallejo, while Fearnley faces Argentina’s Juan Manuel Cerundolo with the likely prospect of world number one Jannik Sinner waiting in round two.
Samuel, meanwhile, will face Australia’s eighth seed Alex de Minaur.
Raducanu has been handed a tricky first-round test against Argentine clay-court specialist Solana Sierra as Boulter starts against American wildcard Akasha Urhobo, and 25-year-old Jones plays Brazil’s 2023 semi-finalist Beatriz Haddad Maia.