Three-time champion Clijsters announced earlier in the year this tournament would be her last and, although she still has doubles to come, there will be no fairytale fourth singles title.
Robson, 18, had never won more than one match at a grand slam before but found her best form on the biggest stage to triumph 7-6 (7/4) 7-6 (7/5) and reach the third round, where she will play ninth seed Li Na.
Clijsters has a remarkable record at Flushing Meadows, winning the title on her three most recent appearances in 2005, 2009 and 2010.
A combination of injury and a previous spell in retirement prevented her from playing in the intervening years and her last defeat at the tournament came in the final in 2003, when Robson was nine.
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Although Robson has played on Centre Court at Wimbledon a number of times, this was her first experience of Arthur Ashe Stadium, the sport's biggest arena, and, perhaps not surprisingly, she began a little nervously.
Clijsters hit her stride from the start and was dictating play, breaking the Robson serve in the fourth game and establishing a 5-2 lead.
But the British number three was beginning to find her range and, with Clijsters serving for the set at 5-3, she pounced, forcing her opponent to net a backhand on break point.
There was still plenty of work to do as the Belgian piled on the pressure in the 12th game and three times brought up set point, but on each occasion Robson's big serve got her out of trouble and into a tie-break they went.
Clijsters moved ahead 4-3 but Robson chose the perfect time to pull out her two best returns of the match and confidently served out the set.
The world number 89 loves a big stage and has the power to trouble even the very top players, but had never previously taken a set off such an illustrious opponent.
She wobbled in the third game of the second set, sending down consecutive double faults to be broken, but then hit straight back when Clijsters too double-faulted.
Robson, watched by new coach Zeljko Krajan, had two chances to move 4-2 ahead but could not take them and, although the youngster was in the ascendancy, Clijsters was fighting for all she was worth.
The 29-year-old held for 5-5 after a long game but in her next service game she found herself down two match points.
Robson was on the verge but she could not convert as Clijsters powered away a drive volley and then sent down an ace to set up a second tie-break.
It was tight and tense but Robson nailed a forehand down the line to bring up a third match point and this time she took it with a serve that Clijsters could only scramble back over the baseline.
Robson relished the occasion and said: "I've always loved playing on a big stage. I think I thrive in a good atmosphere and I feel like I play better on big courts when I have a lot of support. It's easier when you have less pressure on you.
"I really enjoyed myself out there. I was just trying not to sing at the change of ends. I managed to sing along once to the Taylor Swift song. I couldn't help myself."
Robson, who next faces compatriot Heather Watson's conqueror and ninth seed Li Na, praised Clijsters on court after the match as a great role model, but she kept all thoughts of what a win would mean for her opponent out of her mind.
She said: "I didn't think about that during the whole match at all. If I thought about that, I would have been a lot more nervous for sure.
"It's definitely disappointing to see her retire because she's such a great addition to the women's game.
"She's always been someone that I've looked up to since I started on the tour. She's always been incredibly nice to be around. I think we're all going to miss her."
Robson has twice pushed Maria Sharapova hard at Wimbledon, most recently in the second round of the Olympics a month ago, and she was delighted to finish off the job this time.
The Londoner said: "In the first-set tie-break I was thinking 'I've had a few tough tie-breaks and lost them, so it would be pretty nice to win this one.'
"But I think every match I play against a tough opponent, like Sharapova, I get more experience from it and I learn a lot, and that's what I tried to bring on court today.
"And I thought I played really solid in both tie-breaks. I think previously I just sort of slipped up a little bit in the key moments. Today I managed to stick with it.
"That's what I'm really happy with today, that I managed to keep it up for the whole match."
Clijsters achieved one of the most remarkable grand slam victories of all time in 2009 only a month after ending two and a half years in retirement, and she struggled to contain her emotions as she contemplated the end.
She said: "One of my biggest dreams came true in 2005 winning here. Every time I came back I was so inspired by the energy on this court.
"I've played some of my best tennis and my best matches here. This feels like the perfect place to retire, I just wish it wasn't today.
"Laura played extremely well today. I fought, I gave it my all and it wasn't good enough. Since I retired the first time it's been a great adventure for my team and my family and it's all been worth it, but I do look forward to the next part of my life."
Clijsters also commented on Robson's comments about what a great role model Clijsters has been, and the heartfelt reaction to her retirement from within the tennis world.
Clijsters added: "It does something to you when you hear other players talk about me like that. Obviously in these two rounds that I've played here, I've played players that I spoke to and they said that I inspired them.
"That's a great feeling, because I was once in that situation as well. You think about those kind of things now. Now that I'm almost completely finished, you think about when I first stepped on tour and met Steffi Graf and Monica Seles.
"It's been an incredible journey and a lot of dreams for me have come true because of tennis.
"As a little girl, I got tennis racquets under the (Christmas) tree and outfits of Steffi Graf and Monica Seles and I would want to wear them to bed I was so excited.
"So for me to have been able to have been a part of women's tennis, and on top of women's tennis for so many years...you don't think about it when you're in it.
"Now that I think about it, it's been a crazy rollercoaster at times. All of a sudden when you're 15, you get thrown in the spotlight, you go through puberty in the spotlight, you have your first boyfriend in the spotlight, everything.
"It's not just the tennis side of things that you think about now, it's about life. We've had a lot of things happen in these last 15 years that I've been on tour.
"I'm able to look back at them and I'm very happy with the progress that I've made."
Robson, who reached the third round at a grand slam for the first time, next faces ninth seed Li Na, who recovered from 4-0 down in the second set to beat Australia's Casey Dellacqua 6-4 6-4.
There were more straightforward wins for world number one Victoria Azarenka, third seed Maria Sharapova and defending champion Sam Stosur.
Azarenka opened proceedings on Arthur Ashe with a 6-2 6-2 victory over Belgium's Kirsten Flipkens that was not quite as one-sided as the scoreline suggested but nevertheless very comfortable.
Sharapova was detained for only 54 minutes under the lights by Spain's Lourdes Dominguez Lino, the Russian triumphing 6-0 6-1, while Stosur defeated Romania's Edina Gallovits-Hall 6-3 6-0.
Sharapova next meets 21-year-old American Mallory Burdette, a 6-2 6-4 winner over Czech Lucie Hradecka.
The 21-year-old studies psychology at Stanford and, as an amateur, cannot pick up the USD65,000 for reaching the third round.
Burdette said: "I have already checked the amateur box so you can't go back once the tournament starts. It's done."
Fifth seed Petra Kvitova came through 6-4 6-3 against France's Alize Cornet, while there were also wins for Lucie Safarova, Nadia Petrova, Varvara Lepchenko, Zheng Jie and Marion Bartoli, but seeds Anastasia Pavlyuchenkova and Yanina Wickmayer lost.