Men's second seed Ben Shelton also advanced, as did women's top seed Jessica Pegula and No. 2 Elina Svitolina.
Top-seeded Minaur was always likely to prove too much for 17-year-old qualifier Hewitt - son of Aussie great Lleyton Hewitt - who had only captured his first ATP Tour victory in Tuesday's first round.
The first set threatened to become a demolition, as an efficient De Minaur repeatedly broke Hewitt for a 5-0 lead.
But Hewitt found his feet, breaking back in the sixth game as De Minaur found the net, and then held serve, before succumbing 6-2.
Cheered on by a crowd including his famous father, Hewitt started the second set brightly.
But he was broken again on his second service game, and could not reel back in De Minaur, who warmly embraced the youngster at the net after taking the match.
It was the first head-to-head ATP Tour match between the two men, who have known each other for years, with Hewitt's father captaining Australia's Davis Cup team.
"It was definitely weird," said De Minaur. "I knew I was going to play him on the Tour some day, but I didn't actually expect it to be this week.
"It was an amazing feeling. Even all the lead up, seeing all the incredible pictures that we have together and how we've both grown up.
"I'm glad I managed to get through, but I'm sure we're going to have many battles in the years to come."
De Minaur will play American Brandon Nakashima, who upset Czech seventh seed Jakub Mensik 7-6 (5), 3-6, 6-4 to reach the Washington quarter-finals for a second year running.
US home hope Shelton advanced with a gritty 6-7 (3), 6-3, 6-4, victory over France's Ugo Humbert. After handing Humbert the first set with a double fault on set point, Shelton dominated the second.
But the third set was a tense affair and after seizing a break for a 5-4 lead Shelton had to fend off a break point before polishing off the match with a sharply angled serve that Humbert smacked into the net.
"For me, it was all about heart tonight," Shelton said after lining up a meeting with Chile's Alejandro Tabilo, a 6-3, 7-6 (6) winner over France's Terence Atmane.
Check out the match summary here.
Women's top seed Jessica Pegula overcame an early scare to defeat Poland's Magdalena Frech 3-6, 6-3, 6-0 and book a quarter-final berth.
Frech took advantage of Pegula's wayward serve to win the first set before the American upped her game.
Pegula said she started the match "just a little off" after receiving a first-round bye.
"It's always just a little bit of trying to figure it out," she said.
"I wasn't quite attacking the ball, and I felt like once I did that, then everything else kind of started to click together.
"And then just trying to figure out my serve. I knew I could serve better than that," added the American after winning 12 of the last 15 games.
Pegula next plays fifth-seeded Anna Kalinskaya, who saved two match points to beat Indonesia's Janice Tjen 1-6, 6-3, 7-6 (7).
Ukraine's Svitolina booked her quarter-final berth with a 7-6 (4), 6-4 victory over Uzbekistan's Polina Kudermetova. She'll face rising Philippines star Alexandra Eala for a place in the semi-finals.