The Australian Open champion just needs to win her quarterfinal match on Friday to snatch the top spot from Caroline Wozniacki of Denmark.
Meanwhile, Andrea Petkovic of Germany, Yanina Wickmayer of Belgium and Sofia Arvidsson of Sweden advanced to the second round.
Clijsters got off to a sluggish start, dropping serve twice to give Barrois a 4-0 lead.
The top-seeded Belgian fought back to cut the deficit to 4-3, but Barrois held her nerves, converting her first set point when Clijsters sent a backhand volley into the net.
Clijsters hit a backhand winner crosscourt in the next set for a 3-1 lead. She broke Barrois with a backhand winner for a 4-2 lead, clinching the second set when a forehand from Barrois sailed long.
Clijsters was in complete control in the final set, although she had to save three break points in the third game.
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