Kozlov is joined in the quarterfinals by fellow 15-year-old Francis Tiafoe, a 6-1, 7-5 winner over Burundi's Guy Orly Iradukunda, and wild card Deiton Baughman, who defeated his second seed of the week, No. 10 Lucas Miedler of Austria, 7-6(8), 6-3.
Baughman, who reached the quarterfinals of the Orange Bowl last year, said he is comfortable on the green clay, despite growing up in Southern California.
"When I was younger, my dad made sure I played on it a lot, as much as possible," said the 17-year-old, who trains at the IMG Bollettieri Academy. "Over the last year or so, I've been training on it a lot, especially here at IMG, and it suits my game a lot."
Baughman saved nine set points in the opening set, serving for it at 5-4, 40-0 and 6-5, 40-0.
"He had a couple more set points after I got it back to deuce," said Baughman. "I just played really well and I feel he got a little nervous, and in the tiebreaker I played well on the big points."
Baughman got an early break in the second set, and finished strong.
"Overall it was a clean match," said Baughman. "I wouldn't say it's my best of the tournament, but it definitely was a clean match."
Baughman will play Tiafoe, with Kozlov and Rublev also in the bottom half of the draw.
"Last time we played it was for third and fourth place for Nike Junior Tour," said Baughman. "I was like 14 and he was like 9," Baughman joked. "I beat him up pretty good, but he's been playing some good tennis since then and it should be fun. I'm looking forward to the opportunity."
The top half of the boys draw features the only remaining qualifier, with Matias Zukas of Argentina going against No. 12 seed Naoki Nakagawa of Japan. Unseeded Fajing Sun of China will play No. 9 seed Kamil Majchrzak of Poland, who beat No. 5 seed Michael Mmoh 6-4, 6-3.