In a high-quality match on Arthur Ashe Stadium, Robson got the better of the 23rd seed in two tie-breaks to seal her progress to the third round of the final Grand Slam of the year.
Three-times champion Clijsters, who is due to retire after the tournament, led 5-3 in the first set and had three set points but was outplayed by the 18-year-old, ranked 89.
Clijsters, the winner at Flushing Meadows in 2005, 2009 and 2010, had three set points at 6-5 on the Robson serve but the Briton fought back to win the first tiebreak 7-4.
The Belgian broke to lead 2-1 in the second set but Robson broke back immediately and though Clijsters saved two match points to force a tiebreak, Robson held her nerve to take it 7-5.
"It was such a good atmosphere," Robson said. "I didn't have any expectations because she's so experienced.
"I've had so many tough matches against top players that I knew the level was there, I just had to keep it for the whole match.
"Kim has been such a great player for so many years and a role model for all the younger girls coming up."
Clijsters, who is bidding goodbye to the game for the second time after an initial retirement in 2007, said: "One of my biggest dreams came true in 2005, winning here. Every time I came back I was inspired.
"This completely feels like the perfect place to retire. I just wish it wasn't today but there was no doubt in my mind to pick a tournament to play my last singles match.
"Laura played extremely well. I fought and gave it my all but it just wasn't good enough. I look forward to the next part of my life."
Robson now goes on to meet China's Li Na, who recovered from a mid-match lapse to beat Australian Casey Dellacqua 6-4 6-4.
The ninth seed, who has never been past the quarter-finals in New York, recovered from 4-0 down in the second set to reach the last 32.
Former French Open champion Li edged the first set but Dellacqua, a left-hander ranked 92nd in the world, blasted a series of winners to move clear in the second.
But Li recovered her poise in time to reel off six straight games and set up a clash with Robson.
Former champion Maria Sharapova steamrolled Lourdes Dominguez Lino of Spain 6-0 6-1 in another lopsided night match in Arthur Ashe Stadium.
Sharapova, champion at Flushing Meadows in 2006, needed only 54 minutes to sweep aside the Spaniard and advance to a third-round encounter with American wildcard Mallory Burdette.
The Russian's swift victory followed a similarly quick win by Serena Williams on the main show court the previous evening.
Third-seeded Sharapova hit 29 winners to none from the outclassed Lino.
World number one Victoria Azarenka rolled into the third round with an emphatic 6-2 6-2 victory over Belgian qualifier Kirsten Flipkens.
Azarenka never lost her serve to win the one-sided affair in 65 minutes on a blustery Arthur Ashe Stadium court.
"It was another good performance today," said Azarenka, who lost just one game in her first-round demolition of Russian Alexandra Panova. "It was a little bit difficult with the wind today, to adjust.
"It was blowing a lot from one side. You kind of had to adjust every time you changed sides. When the wind is behind you had to be more patient, sometimes more aggressive, more spin."
Seeking her second Grand Slam title after winning the Australian Open this year, Belarusian Azarenka will face 28th seed Zheng Jie of China in the third round.
Zheng moved in the second round with a 6-3 6-1 win over Magdalena Rybarikova of Slovakia.
Azarenka's punishing groundstrokes were too much for the 133rd-ranked Flipkens, who had advanced to the second round of a major for the first time since Wimbledon in 2010.
Flipkens led 2-1 in the second set but squandered two break points opportunities and the train fell off the rails.
Defending champion Sam Stosur stormed into the third round, crushing Edina Gallovits-Hall of Romania 6-3 6-0.
The Australian, who clinched her only Grand Slam singles title in New York last year, dominated world number 135 Gallovits-Hallas as she wrapped up the match in exactly one hour.
Gallovits-Hall had qualified for the main draw but was outgunned from the start by the power of a confident Stosur.
The Australian ripped through the second set to clinch a convincing victory and will next play American Varvara Lepchenko.
Uzbek-born American Lepchenko, seeded 31, beat Anastasia Rodionova of Australia 6-2 6-2 to secure her progress.
Czech Petra Kvitova, the fifth seed, had 29 unforced errors and only 21 winners but managed to subdue France's Alize Cornet 6-4 6-3 to set up a third-round match against Pauline Parmentier of France.
Parmentier downed 25th seed Yanina Wickmayer of Belgium 7-6(5) 6-3 to book her place in the third round.
Eleventh seed Marion Bartoli progressed despite a second set to forget in a 6-2 1-6 7-5 win over Switzerland's Romina Oprandi and fellow Frenchwoman Kristina Mladenovic shocked 17th seed Anastasia Pavlyuchenkova of Russia 6-1 6-2.
Fifteenth seed Lucie Safarova and 19th seed Nadia Petrova will meet in the third round after winning through their third round contests.
Safarova had to work hard to defeat Aleksandra Wozniak 6-3 4-6 6-2 but Petrova had things much easier in a 6-1 6-1 romp against Romania's Simona Halep.