Raducanu won the first set in 38 minutes after successfully converting her only break point in the fifth game.
She then immediately broke Osaka's serve at the beginning of the second set and after saving a break point in the fourth game, broke her serve for a second time to take a 4-1 lead.
With Japan's Osaka serving to stay in the match, Raducanu could not convert her first match point in the seventh game.
But after spurning a second match point and saving a break point, Raducanu completed the victory at the third attempt, winning the second set 6-2.
Speaking to Sky Sports after the match, she said: "I thought it was going to be a really difficult match.
"Naomi's won four slams, she's been world number one, won Masters. She's so dangerous and on the hard courts, I think she's particularly comfortable.
"I knew I was going to have to play well and manage my own service games, which I'm really proud of how I did. I was making some inroads in her service game after I got used to it a little bit.
"I'm really pleased with how I handled the ball speed and conditions here in DC."
Raducanu is due to be in women's doubles action later on Thursday, partnering Elena Rybakina against Giuliana Olmos and Aldila Sutjiadi in the quarter-finals.