Van Uytvanck is ranked 100 places higher than world No 227 Moore but it was the British No 4 who progressed with a 6-3, 6-2 victory.
Moore will earn £50,000 for playing in round two, where she will meet the 13th- seeded Russian Svetlana Kuznetsova, who beat Caroline Wozniacki in her first-round match.
Moore, 23, lost in the first round at both Eastbourne and Birmingham this month but she had shown good form prior to that, winning 13 out of 16 matches.
Born in Hong Kong, Moore was handed a wild card into the main draw at the All England Club and she justified her place, breaking Van Uytvanck’s serve three times to seal a comfortable win.
“It’s amazing. Who would have thought before the grass season that I’d be where I am right now?” Moore said.
“I think I’ve definitely worked really hard and I’ve definitely put in a lot of graft and I think it’s paying off finally. I’m just really happy.”
Former world No 1 Wozniacki will drop out of the top 50 after her straight-sets by Kuznetsova.
World No 14 Kuznetsova dispatched the Dane 7-5, 6-4 on Centre Court yesterday, to book that second-round clash with Moore.
Wozniacki rose to No 1 in the world in 2010, managing to hold the top spot for 67 weeks.
But the 25-year-old has been unable to maintain that stellar form, and will doubtless view this latest setback as a big f rustration.
Meanwhile, Judy Murray has backed Laura Robson to handle the culture shock of dropping off the main tour as she bids to rebuild her career.
The former British No 1 loses her protected ranking of 58th after Wimbledon, having been afforded that comfort as she returned from a long-term injury absence.
It has meant Robson has been able to compete against, and spend time among, the world’s leading players, but the privilege runs out now.
She reached a career-high ranking of 27th in July 2013, immediately after Wimbledon, but currently stands 283rd, and the one-time Wimbledon girls’ champion finds herself at a career crossroads.
Murray is convinced Robson will throw herself headlong into the challenge of playing minor events on the International Tennis Federation (ITF) tour, a step below the Women’s Tennis Association (WTA) tournaments at which Robson has been accustomed to competing.
Events with total prize-money of 25,000 US dollars and 50,000 US dollars await her, in the short term at least, as Robson bids to climb back up the rankings after almost two years of inactivity triggered by a wrist injury.
“I think she’ll be okay with it,” said former Fed Cup captain Murray. “I think she’s addressed these two years incredibly well, for a young person.
“It’s not easy to rehab a long-term injury, particularly when you’re very young, because you think you’re invincible.
“I think she’ll manage that fine and having played in a lot of the big events in the last few months and realising she’s not quite ready for that yet... I think she’ll know, she’s smart, she’ll know that’s what she has to do and I’m sure she’ll apply herself to doing that.”
Robson, 22, was beaten in the first round at Wimbledon on Monday by Germany’s Australian Open champion Angelique Kerber.
She has won just one main-draw WTA match all season, and will be determined to be more competitive in the future – something Murray thinks will only come through winning lower-tier matches and building confidence.
“She’ll enter the 50Ks, the 25Ks, that’d be the best place to start,” said Murray, speaking on behalf of HSBC during activity on their Court 20 area alongside the famous Wimbledon queue.
“The key is to play lots of matches and get lots of wins, at whatever level. Wins build the confidence and when you’re confident it’s very much easier to keep moving upwards.”
Robson would be likely to receive a Wimbledon wild card next year, should her ranking not guarantee automatic entry, and Murray is certain she will be back.
“I’m sure she will be. If she wants to be, she will be here,” said Murray, mother of Davis Cup winners Jamie and Andy.
“She’ll need to work hard and go the right route and have sensible people around her who really support her because it’s hard, because it’s been a long time for her now and it must be very frustrating.”