Svitolina has endured a disappointing start to the season and there was no upturn in fortunes in Doha, where she was beaten 6-3 6-3 in the second round.
Martic got past Barbora Strycova in the first round in Dubai last week, but the Czech gained revenge with a 4-6 6-1 6-1 victory to move into round two.
Alison Riske, the 13th seed, lost an all-American encounter with Jennifer Brady, while 17th seed Donna Vekic went down in straight sets to Iga Swiatek.
Kiki Bertens had no such trouble, winning 6-2 6-4 against Karolina Muchova to reach the third round, while Elena Rybakina came from a set down to oust Sorana Cirstea.
TEENAGER ANISIMOVA STARTLES SVITOLINA
Svitolina has won just four matches this year and it was 18-year-old Anisimova who inflicted another blow on the world number seven.
The American teenager claimed a first top 10 win of 2020, taking just 64 minutes to sail through in her first appearance in this tournament.
She struck 27 winners and saved all three break points faced and will take on either Swiatek or Svetlana Kuznetsova for a place in the quarter-finals.
RYBAKINA ROARS BACK
Rybakina suffered the agony of a defeat to Simona Halep in the final of the Dubai Tennis Championships on Saturday.
The 20-year-old rising star from Kazakhstan, up to a career-high 17th in the rankings, gave another demonstration of her fighting spirit by roaring back to beat Cirstea 3-6 6-3 6-1.
Rybakina already has 20 victories to her name this season and will be expected to make it 21 when she takes on Alison Van Uytvanck in the second round.
SUAREZ NAVARRO SETS UP BATTLE OF FORMER CHAMPIONS
Carla Suarez Navarro coasted to a 7-5 6-0 first-round victory over Zhang Shuai on day two of the tournament.
The Spaniard's reward will be a tough encounter with Petra Kvitova, two players who have experienced glory in Doha.
Suarez Navarro won the tournament in four years ago and Kvitova was crowned champion in 2018.
Barty collected the tour's top individual year-end award in 2019, having won the French Open and finished that season as the world number one.
And after Sofia Kenin succeeded her in 2020, Barty is back on top this year.
This was the third straight season the Australian superstar claimed the year-end number one ranking, and it saw her land a second major title at Wimbledon. Barty's five WTA titles in 2021 were unsurpassed.
But she has had to share the limelight in recent months with Raducanu, and that was the case again as the 19-year-old Briton was unsurprisingly hailed as the tour's star newcomer.
Raducanu became the first qualifier to win a grand slam when she swept to glory at the US Open, having also impressed in a run to the last 16 at Wimbledon.
The teenager, who has risen from 343rd to 19th in the rankings in the past year, won 10 straight matches without dropping a set from the first qualifying round to the final at Flushing Meadows.
Elsewhere, Roland Garros champion Barbora Krejcikova collected a pair of awards, scooping Most Improved Player and, alongside Olympic, WTA Finals and French Open-winning partner Katerina Siniakova, the Doubles Team of the Year.
Carla Suarez Navarro, who retired in 2020 and was then diagnosed with Hodgkin's Lymphoma, returned to the tour cancer free in 2021 for a farewell season and carried off the Comeback Player of the Year award.
Now 34, Suarez Navarro was diagnosed with Hodgkin lymphoma in September 2020, just months after announcing her intention to retire.
She delayed that retirement and, after being given the all-clear in April 2021, began a farewell tour at that year's French Open that included appearances at Wimbledon, the US Open, the Olympic Games and the Billie Jean King Cup.
Having left competitive tennis behind at the end of the 2021 season, Suarez said she and her partner Olga Garcia were thrilled to announce the pregnancy, with their baby due in June.
Garcia, 30, is a footballer who has played for the Barcelona, Atletico Madrid and Spain women's teams.
They said in a shared statement on Instagram: "A new stage that we really wanted to experience. Let's grow the family! Excited and happy to face this big moment. Very soon we will be one more."
Both were showered with messages of congratulations and love, from the likes of Manchester United goalkeeper David de Gea and tennis names including Caroline Wozniacki, Garbine Muguruza, Paula Badosa and Sara Errani.
Suarez Navarro explained, in a statement reported by Spanish media: "We wanted to announce that I am three months pregnant. It's already 12 weeks of tremendous joy to bring a new life into the world.
"Both Olga Garcia, my partner, and I feel an enormous responsibility, a great pride for having taken this step, and we were excited to share it with all of you.
"This Tuesday, we went to perform the first trimester ultrasound, to verify that the pregnancy is progressing on the right track. Fortunately, the medical team has confirmed that the gestation process is developing completely normally.
"It is fantastic news for us and fills us with hope for the near future."
The former WTA player, who reached a career-high world ranking of sixth, added: "After my retirement from sports, with both of us still young, we thought it was the ideal time to face a process that fills us with joy and enthusiasm.
"We are very excited with what we are going through and with all the experiences that we hope to live soon."
Former world number six Suarez Navarro last week withdrew from the US Open due to health reasons.
The 31-year-old Spaniard announced last December that the 2020 season would be her last on the WTA Tour.
In a video posted on her official Twitter account on Tuesday, Suarez Navarro said: "Hi everyone, just wanted to let you know that a few days ago I have been diagnosed with lymphoma. It's Hodgkin lymphoma. I'll need to go through six months of chemotherapy.
"I'm fine and calm at the moment, willing to face whatever comes."
The accompanying tweet read: "I have to face a complicated reality. It is time to accept it and try to move forward relying on medical advice. Always with positivity in the face of adversity.
"Patience and self-belief guided me through my career. Not the easiest rival to deal with. I'll need my true best."
Barty – champion in Paris back in 2019 – had to take a medical time out during her first-round tie with Bernarda Pera on Tuesday.
The Australian world number one struggled with a hip issue, yet still progressed with a 6-4 3-6 6-2 win.
"It's going to be a little bit tough this week. I think over the weekend we had a bit of a flare-up through my left hip, which obviously just needed a bit of help today, needed some assistance to try and release it off as best I could," the 25-year-old said in her post-match news conference.
"I'm not going to hide behind the fact I'm not quite 100 per cent but I can guarantee that I will go out there with the right attitude every day and be really accepting of that and give it a crack no matter what.
"We're confident in that we can get my body to a point where I'm able to compete. We're here, we're fighting, we're in with a chance, and that's all we can do."
Magda Linette is next up for Barty on Thursday, the Polish player defeating Chloe Paquet in straight sets.
COCO CARRIES ON HER FORM AS SEEDS REACH ROUND TWO
Fresh from her triumph in both the singles and doubles in Parma, and a run to the Internazionali BNL d'Italia semi-finals, Coco Gauff started her campaign with a tough win over qualifier Aleksandra Krunic.
Gauff, the 24th seed, certainly had to do it the hard way, with the 17-year-old clawing back four set points in the opener, but some brilliant first serves – she landed 71 per cent of them – got her through in straight sets.
It brought up Gauff's first win at a grand slam as a seed. She is ranked at 25th in the world, making her the youngest American woman to make it into the top 25 since Serena Williams in 1998.
Wang Qiang will face Gauff in round two in what is a repeat of the Parma final.
Meanwhile, three other big names also made it through. Former world number one and 2017 French Open semi-finalist Karolina Pliskova overcame her close friend Donna Vekic 7-5 6-4, while fellow Czech Karolina Muchova came from behind to beat Andrea Petkovic 1-6 6-3 6-4.
With Petkovic's exit, there are no female players from Germany in the second round at Roland Garros for the first time since 1958.
Fifth seed Elina Svitolina is also through to round two after coming back from a break down in the second set to beat teenager Oceane Babel 6-2 7-5.
"It's a first time for her to play here in the main draw of the grand slam and to face the sixth player in the world it's extremely tough," Svitolina said of her young opponent.
"I think she dealt very well with it and she went for her shots in the second set and made life very difficult for me."
SUAREZ NAVARRO MAKES COMEBACK
Carla Suarez Navarro overcame Hodgkin lymphoma earlier this year, after revealing last September that she had been diagnosed with the rare cancer.
Suarez Navarro won that battle and is now cancer free, but in what she said will be her last appearance at Roland Garros, she slipped to a defeat to 2018 finalist Sloane Stephens.
The Spaniard took the first set 6-3 and forced a tiebreaker in the second, but it was Stephens who edged it before taking that momentum into the decider to claim a 3-6 7-6 (7-4) 6-4 win.
Nadal will start his quest to win the Paris grand slam a staggering 14th time with a first-round encounter against Australian Alexei Popyrin next week.
Defending champion Nadal, the third seed, is in the same half of the draw as fellow all-time greats Djokovic and Federer, who could face the Serbian world number one in the last eight.
Top seed Djokovic, who is two major titles shy of the record of 20 held by Federer and Nadal, will take on Tennys Sandgren in the first round.
Swiss great Federer will come up against a qualifier in round one at Roland Garros, while two-time runner-up Dominic Thiem is up against Pablo Andujar.
Pole Swiatek claimed her maiden grand slam title at the French Open last year and takes on her close friend Kaja Juvan in the first round.
World number one Barty, who did not travel to Paris to defend her title in 2020 due to the coronavirus pandemic, has will make her return at the clay-court major against Bernarda Pera.
Serena Williams comes up against Irina-Camelia Begu, while last year's runner-up Sofia Kenin must do battle with the 2017 champion Jelena Ostapenko in a standout first-round match.
Carla Suarez Navarro can expect plenty of support when she takes on Sloane Stephens in her first tournament since successfully completing cancer treatment.
The former world number six revealed last September she would undergo six months of chemotherapy after being diagnosed with the rare cancer.
Suarez Navarro vowed at the time to show "positivity in the face of adversity" and the 32-year-old has now announced she is cured.
The 32-year-old posted on social media: "Another step forward. Today I finished my treatment and overcame Hodgkin lymphoma.
"Thanks to all for your warm messages. Every word of support gave me strength during the past few months.
"All my gratitude to healthcare professionals who take care of us every day. I'M CURED!"
Carla Suarez Navarro announced two years ago that the 2020 season would be her last on the WTA Tour.
However, the WTA said she decided to has abandon that plan and would return to action, confirming her entry for the French Open which begins on May 30 in Paris.
Suarez Navarro was diagnosed with Hodgkin Lymphoma in September last year, just months after announcing her intention to retire.
She delayed her retirement and, after being given the all-clear in April, began a farewell tour at the French Open that will end with the year's final grand slam, the US Open, which starts in August.
Having lost to Sloane Stephens at Roland Garros, Suarez Navarro took a set off world number one Barty on Centre Court, before eventually suffering a 6-1 6-7 (1-7) 6-1 defeat.
Barty and the Wimbledon crowd gave her a standing ovation as Suarez Navarro left the court, ensuring it was a day she will never forget.
"Wimbledon make me a really good gift," Suarez Navarro told a media conference.
"I cannot ask for anything else better than this day, one of my last match here, against Ash, world number one, Centre Court, with the roof, was amazing.
"Today I am the most happy player in the tournament. The crowd was amazing, was fantastic, I'm in love with this tournament, this court and this crowd."
Barty said of the former world number six: "She's a hell of a competitor, hell of a fighter, it was a privilege to be able to share that moment, share that court with her today.
"I hadn't had the opportunity to play Carla and it was really special to experience what she can bring from the other side of the court.
"All credit goes to her for her resilience and her nature as a competitor to be able to come back from the adversity that she has and to be able to have that moment with her was nothing shy of remarkable.
"I just said to her it was a pleasure to share the court with you, she's an exceptional person, a great fighter, a great competitor and very well respected in the locker room.
"She's going to be sorely missed, I just wanted to give her the appreciation that she thoroughly deserves, I didn't know what else to do, I wanted to give her a hug and just say congratulations on an exceptional, remarkable career."
Barty will face Anna Blinkova in the second round.
