Kasatkina of Russia, the 26th seed, led 5-3 in the second set and held a total of four set points as she looked to push the match to a decider.
But the 25-year-old Romanian saved her best tennis to turn the match around for a straight sets victory.
"It was a good match. I think I played really well the whole match, even if she has the 5-3 in the second set," Halep said afterward. "I felt that I was pushing her back, and also I was rolling the ball pretty well. I think it was a very high standard of tennis today. Also, she played really well. I mean, she's a very tough opponent when I play her always.
"It's difficult even if I won the first set 6-0, I knew it's gonna be tougher, the second one. I'm really happy with the way I played today."
Halep, the runner-up at Roland Garros in 2014, almost doubled her output of winners (25) to unforced errors (13). She wrapped up the first set in 27 minutes and broke Kasatkina's serve six times.
"It's nice to be in Paris in the fourth round," Halep said on Court Suzanne Lenglen moments after her win.
Halep will next play No. 21 seed Carla Suarez-Navarro of Spain, who was a 6-4, 6-4 winner over 14th-seeded Elena Vesnina of Russia.
Former top-ranked Caroline Wozniacki of Denmark advanced to the to the fourth round, beating American teenager CiCi Bellis 6-2, 2-6, 6-3 when their match resumed after being suspended due to darkness on Friday.
The 11th-seeded Wozniacki's victory over the 18-year-old Bellis, the youngest player in the third round, puts her into the second week of the French Open for the second time
When play was suspended on Friday, Bellis was leading 5-2 in the second set and serving to level the match. She closed out the second set on Saturday and then went a break up early in the third set, before Wozniacki immediately broke back and again in the eighth game to serve out the match.
Cornet had never advanced further than the fourth round in her 45 appearances in a Grand Slam event
"You work all year for moments like this," Cornet said in her on-court interview. "I couldn't dream of a better scenario. I feel so good on this center court."
Garcia, seeded 28th, reached the fourth round of a Grand Slam event for the first time with a 6-4, 4-6, 9-7 victory over Su-Wei Hsieh of Taiwan.