Third-seeded Jankovic shrugged off gusting winds to sweep aside Ai Sugiyama of Japan 6-1, 6-1 before Davenport battled to a 6-4, 5-7, 6-2 win over Taiwan's Chan Yung-Jan.
Jankovic secured victory in 58 minutes after dominating her match with rasping groundstrokes on the showpiece Stadium Court at the Indian Wells Tennis Garden.
"I was very focused on the court and I really tried to do my job out there," Jankovic said.
"I didn't want to let her get into the match. Especially with the wind, you cannot lose your concentration because, as soon as you do that, you start making a lot of errors and lose your rhythm," she said.
Jankovic struggled with her serve in the swirling winds and double-faulted six times.
"I had a different toss every time," the 23-year-old said. "I was trying to find a rhythm but I couldn't."
The Serb, who has reached at least the quarterfinals in all five events she has played this year, broke Sugiyama three times in each set and will next meet Spaniard Anabel Medina Garrigues, who beat Russian Evgeniya Rodina 6-3, 7-5.
Former world No. 1 Davenport, champion at Indian Wells in 1997 and 2000, held off a second-set fightback by Chan to seal victory in just over two hours.
"It was tough," the 31-year-old American said after an error-strewn performance that featured eight aces and nine double-faults. "I really felt like I should have, could have, played a little better and focused. I was up in the second [set] and had a lot of chances and a lot of opportunities that just kind of slipped away.
"Having said that, I'm happy to walk off the court playing the way I did the last four games, taking back control and playing better. Next time, hopefully I can keep my cool a little better and focus in on the task at hand," she said.
Sixth-seeded Bartoli needed eight match points before winning 6-3, 7-5 after leading 5-1 in the second set.
"It's always difficult to close out a match, no matter if you're confident or not," the 23-year-old said. "I made some bad choices in some bad moments and it just turned around."
Australia's Casey Dellacqua advanced when Austria's Tamira Paszek retired hurt because of illness when trailing 6-1, 4-0 and 13th-seeded Francesca Schiavone beat fellow Italian Flavia Pennetta 6-3, 6-4.