Serena Williams said she wants to make her daughter proud after fighting back to reach the French Open third round.

The 36-year-old American came from a set and break down to defeat Australia's Ashleigh Barty 3-6, 6-3, 6-4.

The three-time champion and former world number one, dressed again in her all-black catsuit, is playing her first Slam since winning the 2017 Australian Open while two months pregnant.

However, her stay in Paris almost proved short-lived until she summoned the same fighting spirit which  helped her conquer life-threatening blood clots after giving birth to daughter Alexis Olympia Ohanian in September.

"It's just incredible," said Williams.

"I am so happy to be back here. I lost the first set and thought I must try harder, I must try harder -- and then Serena came out.

"I will be back tomorrow to play doubles with Venus. I don't mind playing singles and doubles -- I will fight my heart out."

Williams said her daughter inspires her to keep winning as she targets Margaret Court's all-time record of 24 majors.

"This is my first Grand Slam back and I want to the best that I can.

"I want to be able to one day tell my daughter that I tried my best." Williams will face Germany's Julia Goerges for a place in the last 16 on Saturday.

If she gets past that challenge, she could face old rival Maria Sharapova for a place in the quarter-finals.

Williams again played in her striking all-black catsuit which has raised some questions over its legality.

But the American insisted it was crucial to help prevent a return of life-threatening blood clots.

"I definitely thought there would be a lot of reactions over the outfit," she said.

"It's not a typical outfit. But what's typical, you knowIJ Who makes the rulesIJ "I feel like in 2018 the world's different. It's important to be yourself and to bend things and to try something new and to try something fresh.

"However, my outfit is actually functional for me, because I had a lot of health issues."

Williams had cruised through her opening round match against Kristyna Pliskova in what was a record 317th win at the Slams.

But Thursday was a tougher proposition for the American who was playing just her sixth match of 2018.

Barty wrapped up the first set when the former world number one netted a weary-looking forehand, the 12th unforced error of the opener.

Williams hit just three winners while 22-year-old Barty could afford to bide her time, just two winners and four unforced errors coming from her side.

She was a break down again after dropping serve in the opening game of the second set.

But just as the half-full crowd, enjoying the late evening sunshine on Court Philippe Chatrier anticipated a quick kill, Williams came storming back, reeling off four straight games for 4-1.

A fifth ace of the contest won her the set.

A break in the third game of the decider proved crucial and Williams claimed victory on a second match point with a sweeping, running forehand.