Gauff is 24 years younger than Williams, who along with sister Serena inspired her to pick up a racket, but was composure personified in her first main draw match at a grand slam, winning 6-4 6-4.
The teenager became the youngest player to qualify at Wimbledon in the Open era on Thursday having sat an online science exam the night before.
Gauff stated her goal is to win the title, saying: "I want to be the greatest. My dad told me that I could do this when I was eight. Obviously you never believe it. I'm still not 100 per cent confident. But you have to just say things. You never know what happens."
Second seed Osaka's last match before arriving at the All England Club was a comfortable defeat by Putintseva at the Nature Valley Classic in Birmingham and again the Kazakh proved too consistent.
Osaka led by an early break but could not hold onto her advantage and Putintseva celebrated a 7-6 (4) 6-2 victory and one of the biggest successes of her career.
It was clear how disappointed Osaka was by the defeat in her press conference as she spoke barely louder than a whisper and gave very brief answers before cutting it short, saying to the moderator: "Can I leave? I feel like I'm about to cry."
The 21-year-old has not reached the final of any tournament since winning her second successive grand-slam title at the Australian Open in January and climbing to world number one.
There was also a first-round loss for 16th seed and French Open finalist Marketa Vondrousova, who struggled with a hand injury during a 6-4 6-4 defeat by American Madison Brengle.
Tenth seed Aryna Sabalenka, meanwhile, continued to struggle after her breakthrough 2018, losing 6-2 6-4 to 2017 semi-finalist Magdalena Rybarikova, while Nottingham champion Caroline Garcia and last year's quarter-finalist Daria Kasatkina were also knocked out.
Third seed Karolina Pliskova had to save three set points in the second set against China's Zhu Lin before coming through 6-2 7-6 (4).
Pliskova is among the big favourites to win the title and arrived at the All England Club having blitzed the field to lift the trophy at Eastbourne on Saturday.
The Czech, who has only once made it past the second round here, said: "It's tough to compare with the week in Eastbourne. I think it was quite normal. Of course, it went a little bit longer, and I almost dropped a set. It's still almost. I played two sets, that's fine."
Seventh seed Simona Halep was given a tough battle by Belarusian Aliaksandra Sasnovich before battling to a 6-4 7-5 victory.
Halep, who has made the quarter-finals twice and the semi-finals once, trailed 5-2 in the second set and had to save a set point at 5-4 before sealing victory with a run of five games in a row.
Halep suffered a nasty-looking slip during the first set and called for the trainer prior to the start of the second.
The Romanian said: "It's OK in this moment. But I need to do the treatment. I feel it a little bit. I think it's a little bit stretched, the muscle behind the knee, and also the Achilles. Hopefully it's nothing dangerous and I can recover until the next match."
Eighth seed Elina Svitolina, Caroline Wozniacki, Victoria Azarenka and Madison Keys all moved safely through to round two but last year's junior champion, Pole Iga Swiatek, was beaten by Viktorija Golubic.