Third-seeded Jankovic, the runner-up in 2005 and a semifinalist the last two years, lost her first match with Kaia Kanepi of Estonia 6-2, 7-5. It was Kanepi's first win over a top five player.
After Serena broke Zheng in the 10th game, her third straight win over Zheng was assured.
"I think the intensity was high today, but my first serve could have been better," Serena said. "I had to rely a lot on my second serve. I definitely could do better technically."
Venus started aggressively against Cornet then began to dominate when she reined herself in and began to make less errors.
"I made more errors than yesterday," Venus said. "I rushed a little bit in the first set and then realized it in the second and started pulling back a little bit on my shots."
Win those matches and the sisters will play each other for the first time since the WTA Championships in November, when Venus won en route to the title. Neither sister has won the Dubai title.
In the other quarterfinals, fifth-seeded Vera Zvonareva will play Virginie Razzano of France, and Kanepi will meet qualifier Elena Vesnina of Russia, guaranteeing one unseeded player into the semifinals.
Jankovic, down 6-2, 4-0, pulled level in the second set at 5-5 but Kanepi broke in the 12th game for her best win.
"I am really delighted. It was my tactic to mix it up and it worked," Kanepi said.
"I did get a bit tight and started thinking too much when I was 4-0 up in the second, but such things happen when you have the opportunity to beat such a top player. Thankfully, I managed to pull it back," she said.
Jankovic, ranked No. 1 until last month, was embarrassed.
"This was the worst match of my career. It was a horrible day. I kept framing the ball, kept making unforced errors and could not put two balls together on the court. I didn't move properly, and I didn't see the ball properly," she said.
"She didn't have to do too much. Basically, I beat myself. I don't know what happened out there. I am ashamed of this performance," she said.
Zvonareva advanced when Marion Bartoli quit while trailing 5-2 because of painful blisters in her feet.
Dominika Cibulkova of Slovakia also retired after feeling sick from the heat, with Vesnina up 4-6, 6-1, 4-0.