Watson raced through the first set against American Christine McHale to win 6-3 in just 39 minutes.
However McHale fought back in the second to break twice, winning 6-1 and forcing a decider.
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In the final set Watson broke twice to overcome McHale 6-1and rech the semi-final, in which she will face China's Xiyu Wang.
However, it was a different story for her compatriot Edmund, who was defeated in straight sets by Taylor Fritz, 6-4 6-3.
Elsewhere Rafa Nadal and John Isner also made the semi-finals, while Grigor Dimitrov defeated Stan Wawrinka to secure his place.
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