Top seed Williams won the first set 6-4 and was a tie-break away from defeating the 26th seed when they came off.
After a two-and-a-half hour delay, Williams converted a fifth match point to win 6-4 7-6 (12-10) in Paris.
The American, 34, aiming for her 22nd Grand Slam title, was joined in the next round by older sister Venus.
Venus’s match against France’s Alize Cornet was stopped at 2-1 because of the rain, but the 35-year-old came back to seal a 7-6 (7-5) 1-6 6-0 win.
The American, who turns 36 next month, was playing in the third round for the first time since 2010 and has not made the quarter-finals in a decade.
Her sister Serena had dropped just five games in her first two matches at Roland Garros, but faced a tougher test against the French number one.
After the match finally resumed on Court Philippe Chatrier, Mladenovic opened a 5-2 lead in the tie-break – and spurned a set point at 9-8 – before two wayward forehands saw Williams through.
“It was very difficult, she played really, really well,” Serena said. “She forced me to bring out my best tennis.”
She will play in the fourth round against Ukraine’s 18th seed Elina Svitolina, who won 6-4 6-4 against 2008 champion Ana Ivanovic.