Williams was broken twice during the second set Tuesday, including when she was serving for the match at 5-4. At 5-3 with Zakopalova serving, she had five match points but couldn't end it.
"I just played horrendous," said Williams, who is seeded second. "I was a little nervous because I hadn't won a match on clay all year."
Zakopalova saved three more match points before holding to 5-3 in the third set and then broke Williams in the next game to get back on serve.
"I think Serena will be playing better and better each round, so it was the best chance to at least play with her or beat her," Zakopalova said. "She's Serena."
Williams began her Serena Slam at the French Open in 2002, winning the first of four straight Grand Slam titles. If she wins at Roland Garros this year, she'll have won three majors in a row after victories at the U.S. Open and the Australian Open. She reached the final at Wimbledon last year but lost to big sister Venus.
"Now hopefully I'll start playing better," the 10-time Grand Slam champion said.
Jankovic dominated her opponent before a two-hour rain delay and then did well enough after it to beat Petra Cetkovska of the Czech Republic 6-2, 6-3.
The Serb, once ranked No. 1, was leading 4-1 when the rain started at Roland Garros. She quickly completed the first set when play resumed and continued to play well on Cetkovska's serve in the second but was broken twice on her own.
"I was controlling the points ... but then we had to stop because of the rain," Jankovic said. "I felt a little bit slow after the rain delay."
While serving for the match, Jankovic again struggled and was forced to save break points before finally winning.
"The serve was all right. I didn't go for too much," said Jankovic, who added the balls were heavier than usual because of the wet weather. "I just tried to have a high percentage."
Jankovic finished last season as the top-ranked player on the women's tour, but the 24-year-old Serb is still looking to win her first Grand Slam singles title after losing in the final of last year's U.S. Open.
Kuznetsova defeated Claire Feuerstein of France 6-1, 6-4. The 2004 U.S. Open champion also dominated before the rain started falling, leading 5-1.
Dementieva overcame a nervy start to beat South African qualifier Chanelle Scheepers 6-4, 6-3 and book her place in the second round.
Making her Roland Garros debut, the 25-year-old Scheepers broke twice to surge to a 3-0 lead before the Russian world No. 4 reeled off five games in a row.
The Olympic champion squandered two set points at 5-3, fluffing an easy shot at the net and sending a groundstroke long, before securing the set two games later with a powerful backhand that her opponent could only hit long.
The Russian, runner-up in Paris in 2004 and ranked 145 places above Scheepers, broke twice for a 5-2 lead in the second and served out the match, wrapping it up in 83 minutes with a backhand down the line.
Information from The Associated Press and Reuters was used in this report.