The 89th-ranked British player won 7-6 (4), 7-6 (5) on Wednesday. Clijsters, seeded 23rd, had won 22 straight matches at Flushing Meadows, where she took the title three times.
Clijsters fought off two match points at 5-6 in the second set but couldn’t pull out the tiebreaker.
The 18-year-old Robson is the youngest player ranked in the top 100.
Clijsters came back from her first retirement and the birth of her daughter to stun the tennis world in winning the 2009 Open.