She was swiftly followed out of Roland Garros by two-time Grand Slam champion Garbiñe Muguruza. The Spaniard, who won here in 2016, has fallen at the first hurdle for the second year in succession.
She was defeated on Sunday 2-6 6-3 6-4 by Kaia Kanepi of Estonia, the oldest player in the women's draw.
The match ended under a drizzle and Muguruza missed a service return on match point, then cracked her racket against the ground.
Kanepi, who turns 37 next month, is ranked 46th and is participating in her 15th French Open. Her best showings were quarter-final appearances in 2008 and 2012.
Jabeur, seen as one of the big threats to world No.1 Iga Swiatek at the claycourt Grand Slam, came to Paris at a career-high ranking and with a tour-leading 17 wins on the surface.
With her win in Madrid and runner-up finishes in Rome and Charleston, it seemed Jabeur would have an easy outing against world No.52 Linette when she made a strong start on Court Philippe Chatrier under overcast skies.
The 27-year-old sixth seed suffered an early break of serve before breezing through the opening set in 37 minutes but Linette raised her game.
Linette, 30, had only defeated Jabeur in the first of their four meetings back in 2013, but she staged an inspired fightback in the second set, saving four breakpoints to level the match in the tiebreak.
As drops of rain fell on the main showcourt, Jabeur's level also dipped and she made her frustration evident by kicking balls away after losing points.
The Pole, who lost to Jabeur in the third round last year, sealed the match when the Tunisian wasted a 40-0 lead in the 12th game and found the net on match point to be broken for the second time in the set.
"I had so many tough matches with Ons and last year here I lost to her in the third round so I knew how difficult it's going to be," Linette said on court.
"She was playing so well all this time. I knew I have to be focused and play every single point and try to make her uncomfortable. I am happy that I just managed to fight for every single point."
"I wanted to go as far as I could in the tournament because I played well on clay in Madrid and in Rome, and it's difficult to take that one in," Jabeur said.
"But that's what sport is like and you need to be smart enough to move forward and get back on court. So maybe it was a good thing to lose today.
"I would rather say this and be really tough with myself than waste all the good energy that I got from Madrid and Rome."
Linette will next play Martina Trevisan of Italy or Briton Harriet Dart.
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