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After the opening three games went with serve, eighth seed Wawrinka broke Robredo's serve in fourth game before closing out the opener 6-3 with two of his 15 winners in the set.
Wawrinka then snapped Robredo's serve in the first game of the second set to yet again move ahead, but this time he couldn't close it out as Robredo broke back in the 10th game to make it 5-5.
Soon the pair were in a tie-break and although 17th seed Robredo went up a mini-break early with a scintillating backhand, Wawrinka rattled off consecutive points before claiming the set 7-3.
There wouldn't be a single break of serve in the third set as the pair again had to be separated by a tie-break.
The pair changed ends for the first time with Robredo 4-2 up. The Spaniard looked like he would be taking the game into a fourth set, but then Wawrinka won four points on the bounce to set up two match points.
Robredo saved the first with a big serve but then netted with a forehand pass to send Wawrinka through to the last eight, where he next plays Djokovic, who breezed past Fabio Fognini earlier on Sunday.