A single break in the first set cost last year's finalist at the All England Club but she engineered a break of her own in the second set to level the match.
Andreescu, the 2019 US Open champion, struck first in the decider, opening up a 3-1 lead, but the Tunisian sixth seed broke back immediately.
The match was halted at 3-2 in the decider as rain returned to Wimbledon, resuming after a prolonged inspection under the roof.
The turning point came in the ninth game when Jabeur, 28, broke her 50th-ranked opponent to love and she sealed the 3-6, 6-3, 6-4 win with an ace.
Jabeur will play two-time champion Petra Kvitova in the last 16.
"I've got to thank the rain a little bit for letting me speak to my coach and have a bit of perspective about the match," she said. "I felt like I didn't play my best tennis."
Jabeur last year became the first Arab woman to reach a Grand Slam women's singles final, also making the final of the US Open.
She said it felt emotional to return to Centre Court after her painful defeat to Elena Rybakina in 2022.
"I love the energy, love how beautiful it is and hopefully I can come back and play more matches here," she said.
