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The Spaniard came out on top 2-6 4-6 7-6 6-4 3-6 to advance through to the round of 16 in a time of just over three hours.
Having found the first break of serve in the opening game of the contest, Robredo powered ahead to take the first set.
The second proved to be a tighter affair, with Janowicz losing serve in the ninth game with the score tied at 4-4 to allow Robredo to edge ahead to build a two-set lead.
A tiebreak was required to separate the two in the third set as Janowicz eventually prevailed to get a foothold on the match, before capitalising on his opponent's lost serve in the fourth to level the scores overall.
Robredo raced into a three-game lead in the decider with Janowicz unable to recover, ensuring that the world number 22 progressed to the next round.