The 24-year-old had led at the midway stage then went into the fourth round trailing Stephanie Kyriacou by two shots.
But a superb closing 65 gave her a 19 under par total of 265 and a career landmark win in the fourth of five majors on the women's tour.
"I became a Star Wars fan like a month ago, I love the sentence 'May the force be with you'," revealed Furue.
"Then on the 15th hole the sentence came up in my mind and I was just keep going on with that quote."
Australia's Kyriacou took second, one shot adrift, with Thailand's Patty Tavatanakit third a further shot behind.
"Breathtaking, like incredible and I'm so speechless" beamed Furue, who becomes Japan's third women's major winner after Hinako Shibuno, winner of the 2019 Women's (British) Open, and Hisako Higuchi, winner of the 1977 LPGA Championship.
"I didn't believe that I can get the major win so I'm so happy right now," she added.
"I feel so honoured to be the winner of a major championship, but I feel like I put every effort to win every tournament. I feel very comfortable with this win."
Reflecting on her last-hole heroics she said: "First shot landed on first cut so it was a little bit difficult to hit on the green.
"I just made that shot and then got an eagle putt so that was really good for me."
Kyriacou was split with emotion having got so close to the win.
"It was a good week. Sucks not to hold the trophy. If you told me I would come second at the start of the week I would've been happy," she said.
Behind the front three came American Lauren Coughlan, two shots shy of Tavatanakit, with Korean Ryu Hae-ran the same distance back to complete the top five.
Coughlan had been out in front for much of the day and reflected: "I held the lead of a major for almost 15 holes today, or at least until hole 15.
"So, no, I can be very happy about it. Again, a little disappointed just how I ended up finishing, but very happy."
