Townsend, ranked 116th, followed up her shock win over reigning Wimbledon champion Simona Halep by beating another Romanian, Sorana Cirstea, 7-5, 6-2.
"I didn't know that many people had my phone number," Townsend said when asked how she coped in the wake of her Halep upset.
The 23-year-old Townsend is enjoying her deepest run, surpassing her third-round showing on debut at the 2014 French Open, and will play 15th seed Bianca Andreescu for a quarter-final spot.
"I made third round before but it was in another country. I got a lot of messages. It was a lot of love. My phone started dialing 911 by itself, I dont know what that was about. I just tried to keep my head on straight."
"I'm going to take it as far as I can," she added. We're going to ride this thing all the way."
Canadian teenager Andreescu continued to make serene progress on her first Flushing Meadows main draw appearance as she brushed past two-time US Open runner-up Caroline Wozniacki 6-4, 6-4.
"I think I played great tennis. I'm really pleased with my performance," said Andreescu, also into the second week of a major for the first time.
The 19-year-old, a winner at Toronto and Indian Wells this season, improved to 30-4 for the year despite missing much of the clay and grass portion with a shoulder injury.
"I think everything is just clicking with me," she said. "I've been through a lot with injuries and have just taken everything I've learned from the past couple of years and brought it into this year."
Belgian 25th seed Elise Mertens swept into the fourth round for a second year running with a 6-3, 6-3 defeat of Germany's Andrea Petkovic.
- 'Future of women's tennis' -
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A refreshed Rafael Nadal is second up during the day session at Arthur Ashe Stadium, with the Spaniard taking on South Korea's 170th-ranked Chung Hyeon.
Second seed Nadal received a walkover in the second round when scheduled opponent Thanasi Kokkinakis withdrew due to injury.
The three-time champion holds a 34-1 record here against players outside the top 50, although Chung made the 2018 Australian Open semi-finals before a lengthy injury lay-off sent his ranking spiralling.
Saturday's primetime clash on Ashe pits 15-year-old American Gauff, who made a fourth-round Wimbledon run in her Grand Slam debut, against 21-year-old Japanese star Osaka, the reigning US and Australian Open champion.
"I definitely think it's the future of women's tennis," 23-time Grand Slam champion Serena Williams said. "And I'm really excited to just be a fan girl and kind of watch."
Even world number one and defending men's champion Novak Djokovic can't wait to see what Gauff and Osaka do.
"Coco is definitely a new superstar of tennis. It's great to see her fighting spirit, her intensity," Djokovic said. "She's only 15. She brings so much maturity in her game for such a young age. It's very impressive.
"Definitely a fan of her and fan of Naomi, as well, what she has done in the last couple of years. It's going to be really interesting because they're both very likable players and girls. It's going to be definitely a great match."
Nick Kyrgios and Andrey Rublev do battle in the nightcap on Ashe in a quarter that is wide open following the early exits made by fourth seed Dominic Thiem and Greece's Stefanos Tsitsipas.
Sixth seed Alexander Zverev, who has come through back-to-back five-set marathons, will fancy his chances of sticking around for week two with Slovenia's Aljaz Bedene standing between him and a place in the last 16.