Jabeur arrived at Roland Garros with the expectation she would be one of the most likely to challenge strong favourite Iga Swiatek, but Linette consigned the No.6 seed to a 3-6 7-6 (7-4) 7-5 defeat on Sunday.
Battling Tunisian Jabeur won her maiden WTA 1000 title at the Madrid Open this month and had reached the final of three of the four tournaments she has played on clay this season.
The 27-year-old also started the Paris Major with a tour-leading 17-3 record on this surface in 2022, but she was unable to get the better of the world No.56.
Linette was broken three times in the first set, but fought back from 3-1 down in a second-set tie-break to take Jabeur the distance.
It appeared a final-set tie-break would be on the cards in the opening match on Court Philippe-Chatrier, but Linette had other ideas.
The Pole, who struck 32 winners, decisively broke for a third time when Jabeur was serving to stay in the match to claim her first top-10 scalp of the season, setting up a second-round meeting with Martina Trevisan or Harriet Dart.
A junior French Open champion in 2011, Jabeur must lick her wounds after 47 unforced errors to fall at the first hurdle.