It looked as though Robson could have been heading for an early exit during the opening set as she fell 5-2 behind within 30 minutes.
However, she regained her composure to break back against her Belgian opponent, before forcing the set into a tie-break.
The 18-year-old Brit stuck with Clijsters up until four points apiece. She then claimed a mini-break to go 5-4 in front, leaving her just two points away from a one-set lead. Robson duly served out to take the advantage.
Both players held their service games throughout the second set, with Robson looking the most likely to secure a breakthrough. She actually had two match points while leading at 6-5, but veteran Clijsters produced two winners, including an ace, to bring about another breaker.
The duo both found it difficult to hold their two service points during the early exchanges of the tie-break, which in turn meant neither was able to build up a substantial lead.
With the scores level at five points each, Robson produced a powerful forehand down the line that left Clijsters flat-footed. She then kept her nerve to claim the next point and win the match at the third time of asking.
Today's loss has marked the end of Clijsters as professional tennis player. She leaves the circuit as a four-time Grand Slam winner.
Meanwhile, Robson will now face China's Li Na in round three.