Charleston, was scheduled to play American qualifier Mallory Burdette, while Venus was to face Varvara Lepchenko for a spot in the quarterfinals. Both matches were rescheduled for Friday.
The rain fell steadily at the Family Circle Tennis Center until early evening when crews were finally able to start drying the large puddles on the gloppy, green clay courts.
Once play began, ninth-seeded Jankovic - the 2007 Family Circle champion - needed just 64 minutes to oust 19-year-old American Jessica Pegula in straight sets. Pegula won the first two professional matches of her career this week before falling to the Serb, 6-0, 6-4.
Third-seeded Samantha Stosur retired in the second set of her match against Canadian qualifier Eugenie Bouchard, trailing 1-6, 0-2.
The ninth-ranked Stosur was out of sorts against the 19-year-old Bouchard from the start. She had her right leg wrapped and spoke with a trainer after losing the first set. She retired two games later.
It was Bouchard's first victory over a top 10 player.
Second-seeded Caroline Wozniacki of Denmark moved on without hitting a groundstroke when Andrea Petkovic of Germany withdrew with a calf injury a few hours before their scheduled match.
In the only other match completed Thursday, Stefanie Voegele of Switzerland defeated 10th-seeded Julia Goerges of Germany, 3-6, 6-1, 6-3.
The long rain delay means the year's first clay-court event has some ground to make up to finish by Sunday. There are still four third-round matches to be played on Friday before the quarterfinals are set.
Serena Williams, who was not sharp despite winning her opening match over Camila Giorgi on Tuesday , will have waited almost three days to play again. Venus defeated 19-year-old Monica Puig in three sets Wednesday night.
The sisters would each need to win twice more to set up an all-Williams semifinal here. They haven't met in a tournament since 2009.
Both have enjoyed championship moments at the Family Circle Cup. Serena won titles in 2008 and 2012. Venus won the crown in 2004 and last September clinched the winning points as the Washington Kastles took the World Team Tennis title on Billie Jean King Court.
Other delayed matches included sixth-seeded Lucie Safarova against 11th-seeded Sorana Cirstea and the all-American encounter between Madison Keys and Bethanie Mattek-Sands.
Cirstea, like Serena Williams, also hasn't played since winning on Tuesday.
Jankovic said it was difficult keeping sharp and loose as the delay stretched through the afternoon. "I wanted to take a nap, but I couldn't," she said. "We joked that if they rained it out, we have a go shopping day."
Instead, Jankovic felt fortunate to move to the quarterfinals when so many other competitors will likely have to play twice Friday. "At the end of the day, I was lucky," she said. "I came out and played quite well."