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The Czech lost out in a first-set tie-break after failing to convert any of the nine break points he earned during the opener.
In the second set he was able to snap the Romanian's serve, racing through 6-1 in just half an hour to level up the match.
It looked like Berdych would break at the start of the third, but Hanescu held firm after seven deuces to edge in front.
The Romanian then had his first break points that would have given him the chance to serve for the set, but he couldn't take them and Berdych punished him by breaking in the next game and going on to take the set 6-4.