The top seed and defending champion produced a clinical display to defeat the 17-year-old Hewitt - son of former world No.1 Lleyton - 6-2 6-3 in just 61 minutes.
De Minaur shared a warm embrace with Hewitt at the net as Lleyton looked on from the stands at Stadium Court.
"It was definitely weird," de Minaur said. "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.
A Wimbledon boys' finalist this month, Hewitt has idolised de Minaur while growing up around the Davis Cup team captained by his father.
"Even all the lead up, seeing all the incredible pictures that we have together and how we've grown up, I'm glad I managed to get through," de Minaur said.
"But I'm sure we're going to have many battles in the years to come."
Australia's world No.6 charged to the net with authority and displayed great hands en route to improving to 29-14 this season.
Despite his defeat, Hewitt is set to soar some 252 spots in the rankings to a fresh career-high of around world No.360 after qualifying in Washington and then winning his maiden ATP match.
Competing for the first time since marrying British WTA star Katie Boulter this month, de Minaur next faces American Brandon Nakashima, who ousted seventh-seed Jakub Mensik 7-6 (7-5) 3-6 6-4 to reach his second consecutive Washington quarter-final.
Nakashima holds a 2-1 edge over the Australian, with de Minaur's lone win against the American also coming in last year's quarter-finals before he went on to lift the title.
Meanwhile, fellow Australian Bernard Tomic's surge through the ATP Tour tournament in Los Cabos, Mexico came to an abrupt halt as he was trounced 6-1 6-0 by British fifth seed Cameron Norrie in the quarter-finals.
The 33-year-old Tomic, ranked No.171, was seeking his ATP semi-final berth since 2018 and had reached the quarters after getting into the main draw as a qualifying lucky loser.
He had hinted he was considering hanging up his racquet before snaring the Lincoln Challenger title earlier this month.