World number 10 Jabeur had lost her previous four meetings with Kvitova, but she emerged victorious on Wednesday with a 6-4 6-4 win to reach the quarter-finals.
Jabeur recovered from 3-1 down in both sets at Ken Rosewall Arena and converted half of her six break points on her way to setting up a meeting with Kontaveit.
The two will resume their friendly rivalry after Kontaveit overcame Romanian qualifier Elena-Gabriela Ruse 6-3 6-1.
Jabeur lost out to Kontaveit for the last WTA Finals spot in November and is now seeking revenge in the final warm-up event ahead of the Australian Open.
"Anett is a great player. We had our moments last year," Jabeur said. "I told her, 'You're obsessed with me, you always follow me, so stop [smiling]'.
"I know the pressure is on me, not really on her. But maybe I can get some payback for last year."
Garbine Muguruza is also through to the last eight in Sydney thanks to a 6-1 7-6 (7-4) win against Ekaterina Alexandrova.
The two-time major champion will face Daria Kasatkina, who beat Elise Mertens 6-3 6-4, for a place in the semi-finals.
Third seed Barbora Krejcikova survived a brief fightback from Jaqueline Cristian to advance 6-1 7-5, meanwhile, and Paula Badosa beat home hope Ajla Tomljanovic 6-3 6-4.
There was disappointment for in-form Elena Rybakina, who thrashed Emma Raducanu on Tuesday but had to withdraw from her clash with Caroline Garcia with a thigh injury.
At the Adelaide International 2, Coco Gauff prevailed 6-3 5-7 6-3 in her battle of the teenagers showdown with Marta Kostyuk.
Gauff is one of five Americans in the quarter-finals along with Madison Brengle, Alison Riske, Lauren Davis and Madison Keys, who beat Tereza Martincova 6-1 6-3 to advance.