Gauff dumped out of Qatar Open, Swiatek, Rybakina through
The fourth seed, who was beaten 6-4, 6-2, was followed out of the Qatar Open by sixth seed Jasmine Paolini and eighth-seeded Russian Ekaterina Alexandrova.
There were no such problems for Iga Swiatek or Elena Rybakina, though, as the top seed and newly-crowned Australian Open champion both cruised into the last 16.
American Gauff, who received a first-round bye, was also knocked out of the tournament in her opening match last year.
Cocciaretto will face Ann Li in the next round after only the third victory of her career against a top-10 ranked opponent.
The world number 57 said she was inspired by her friend and skier Sofia Goggia, who took bronze in the women's downhill event at the Milan-Cortina Winter Olympics at the weekend.
"We always send messages and FaceTime to talk about our sports and how you manage some things," she said of Goggia. "She gives me a lot of advice...
"I love how they (skiers) approach the sport. They have fear of nothing. They're focused a lot and they put themselves after everything."
Former world number three Maria Sakkari saw off Paolini, also 6-4, 6-2, to set up a last-16 tie with Varvara Gracheva.
Former French Open champion Jelena Ostapenko was another 6-4, 6-2 winner over Alexandrova.
Swiatek dropped just eight points in the first set as she brushed aside Indonesia's Janice Tjen 6-0, 6-3 in her first match since losing in the Australian Open last eight to Rybakina.
The six-time Grand Slam winner will play Daria Kasatkina for a place in the quarter-finals.
She has won her last six meetings with the Russian-born Australian without dropping a set.
"She can come to the match feeling that she has nothing to lose," said Swiatek of facing Kasatkina.
"There are different ways the match can go, and I wouldn't say that focusing on the last results gives a lot. I'd rather stay ready for the challenge and not really think about the previous ones."
Rybakina continued her fine form with a 6-2, 6-4 success against Wang Xinyu and will next face another Chinese player in Zheng Qinwen.
World number three Rybakina has not played Zheng since a group-stage loss at the 2024 WTA Finals.
Australian Open semi-finalist Elina Svitolina and rising Canadian star Victoria Mboko also reached the third round.