Tunisia's Jabeur, who is ranked ninth in the world, was beaten by unseeded American Kenin in a gruelling match which lasted two hours and 17 minutes in the hot Rome sunshine.
After reaching the quarter-finals in Madrid Jabeur, who in 2022 became the first Arab and Tunisian player to win a WTA 1000 title, looked to be on the right track.
But she slumped to a poor defeat on Friday, continuing a dreadful season so far in which the 29-year-old has had to deal with a chronic knee injury resurfacing.
Jabeur has a losing record for the year, a far cry from the woman who looked like she had the world at her feet ahead of last year's Wimbledon final, which she lost in straight sets to unfancied Marketa Vondrousova after blasting through a series of Grand Slam champions.
World number 42 Kenin will face either Rebecca Sramkova or Katie Boulter in the third round.
Earlier fifth seed Maria Sakkari swept past qualifier Varvara Gracheva in straight sets, 6-2, 6-2.
Former Wimbledon champion Rybakina was supposed to open her title defence in the Italian capital on Friday against Romania's Irina-Camelia Begu.
Instead, the 24-year-old, who represents Kazakhstan, has been replaced by lucky loser Oceane Dodin, the world number 73 from France who was eliminated in the second round of qualifying.
"I am disappointed to have to withdraw from Rome this year, but unfortunately I do not feel well enough to compete," world number four Rybakina said in a statement.
"I have such good memories from last year and was looking forward to defending my title."
Home hope Lorenzo Musetti also withdrew from the men's tournament with a stomach bug and flu-like symptoms midway through his match with France's Terence Atmane.
World number 29 Musetti was a set and one game down when he gave up the ghost in his second-round match with unseeded Atmane.
"I tried with the little strength that I had," said Musetti.
"The heat and general conditions didn't help me. I didn't have the energy. It hurts because Rome is a very important tournament for me."
The Masters 1000 tournament, which runs until May 19th, has had a series of players drop out, with stars Jannik Sinner and Carlos Alcaraz both withdrawing from the men's draw before the start.
Another Italian in Matteo Berrettini opted not to play in his home town, the former world number six saying on Wednesday that he was not ready to play after initially being picked as a wild card and then being put in the main draw to replace the injured Jiri Lehecka.