ROME -- After her first-career win against a Top 20 player on clay, Naomi Osaka was quick not to get ahead of herself. In one of her best performances of her season, the former No.1 eased past Marta Kostyuk 6-3, 6-2 in the second round of the Internazionali BNL d'Italia on Thursday.

The win came a day after her straight-sets win over France's Clara Burel. It is the first time Osaka has posted back-to-back victories on clay since 2019. 

She will face a familiar face in the third round in No.11 Daria Kasatkina. It will be the first meeting between the two since the 2018 Indian Wells final, which Osaka won 6-3, 6-2 to capture her first Hologic WTA Tour title.

"I'm going into this match knowing that she has way more experience than me on clay," Osaka said. "I feel like I'm still a baby giraffe, like I'm trying to learn my steps.

"I'm just extremely humble. I want to learn from the match I played today and hopefully apply that."

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Osaka and Kasatkina practiced together just two weeks ago at the Mutua Madrid Open. It didn't go well for the four-time Grand Slam champion.

"She smacked me in Madrid real bad," Osaka said. "It was actually embarrassing. I felt like I had to apologize to her after that practice.

"I know she's an amazing player. She moves really well. Overall I'm just really excited. She's so nice and her brother is really kind, too."

Facing down No.20 Kostyuk, Osaka blitzed through the 36-minute opening frame with a remarkably clean set of tennis. She struck six winners to eight unforced errors and took advantage of Kostyuk's errors. The Ukrainian has been in consistently good form this season; she made two WTA 500 finals, including the Porsche Tennis Grand Prix three weeks ago.

But Kostyuk struggled to find her range throughout the match. She served at just 40 percent in the opening set and hit five winners to 15 unforced errors, including five double faults. 

Rain suspended play with Osaka eyeing a break point opportunity with Kostyuk serving down 3-1 in the second set. When play resumed, the four-time major champion showed no signs of slowing down. She broke serve with a laser-like inside-out forehand to lead 4-1 and closed out the 71-minute with a barrage of heavy, precision baseline hitting. 

Osaka finished the match with 19 winners to 12 unforced errors.

"I think this match is apparently a lot of firsts for me, so I'm very happy about that," Osaka said. "I was pretty nervous coming back after the rain delay, just knowing that I really want to capitalize on that break point. Excited that I was able to do that."

"Overall, if I had to give myself a rating, I'd say that I can see a lot of progress from my last match to now."

The incremental improvements have buoyed Osaka's spirits. She left Madrid disappointed by her effort against Liudmila Samsonova, taking a 6-2, 4-6, 7-5 loss. The scoreline was respectable, but Osaka walked off the court feeling she played too passively. She retreated to Mallorca to train and regroup. 

"Honestly, I did a really solid block of just focusing on what I want my clay court tennis to look like," Osaka said. "I watched some videos. I watched Rafa. I watched Alcaraz. I watched Rublev actually which is very inspiring because he's smacking the ball.

"I kind of thought to myself, like, I don't want to have regrets when I leave the court."

Clarity has been the challenge for Osaka on clay. On her favored hard courts, Osaka knows how she wants to unfurl her lightning strikes. She has few doubts about her movement and footwork and her precision hitting isn't derailed by irregular bounces.  

With coach Wim Fissette, she is working on finding that same sense of identity and confidence on clay. She's not as far off as some may think. In 2019, she made the semifinals of Stuttgart and back-to-back quarterfinals in Madrid and Rome. In the modern game, power can excel on clay and Osaka knows that. She just needs a few minor tweaks.

"Sometimes I think that Ostapenko won French Open, so maybe I should just stick to my guns," Osaka said. "To be honest, I don't really try to bang the ball. That's, like, just what happens.

"I think for me, I just want to put more spin on it while rotating it a lot more. I think when I'm finally able to achieve that, it will obviously be quite heavy, so I [think] that will be my clay court tennis."