Third-seeded Stan Wawrinka had just beaten Fabio Fognini on Court Suzanne Lenglen when the rain began to fall.
The 28-year-old Croat joins four other grand slam winners in the last eight but, while nine-times Roland Garros victor Rafa Nadal, holder Novak Djokovic, 2015 champion Stan Wawrinka and Andy Murray make the headlines, Cilic barely gets a mention.
"I played really well at the beginning", Halep, who won the Madrid title and reached the Rome final in the run up to Roland Garros, said courtside.
Despite hitting eight double-faults, Khachanov won all 23 games he served against Inser, saving four break points.
The 28th-seeded Fognini, who comes with a reputation for having a fiery temper, briefly grumbled at the chair umpire moments later. The third-seeded Swiss, a three-time Grand Slam champion, faces 15th-seeded Gael Monfils on Court Phillipe Chatrier.
Former top-ranked Caroline Wozniacki is through to the fourth round of the French Open after beating American teenager CiCi Bellis when their match resumed.
Xu and Canadian Gabriela Dabrowski, the No. 9 seeds, only needed 55 minutes to win 6-1, 6-2 against Madison Brengle and Anna Smith.
Play was suspended on Friday because of darkness with Bellis leading 5-2 in the second set and serving to level the match.
"I grew up on clay, playing half the year on clay".
The 30-year-old Scot made only one unforced error in pocketing the opening set, and although he did drop serve late in the second, he broke back immediately to take complete control of his first meeting with the up-and-coming Russian. I haven't spent too much time on the court and won all the matches quite straightforward.
Murray got past the 29th-seeded del Potro 7-6 (8), 7-5, 6-0 on Saturday at Court Philippe Chatrier in a match that lasted almost 3 hours.
Murray did enough to keep his nose in front throughout, while the giant Argentine was unable to reprise the heroics that saw the pair produce such magic in last year's Rio Olympics gold medal match.
It looked a tricky match on paper for the 2015 champion but he recovered from a break down in the opening set to win 7-6 (7/2) 6-0 6-2.
Andy Murray, Stan Wawrinka and Kei Nishikori were among those to advance in the men's draw, while Simona Halep cruised through to further underline her status as strong favourite among the women.
This time the Scot would make no mistake and crunched it out with his fourth ace of the match.
Del Potro was the 2009 U.S. Open champion.