The Spaniard took the first set on a tie-breaker as Evans continued his impressive form, but the 31-year-old's experience guided him to take the set 7-6.
Robredo took confidence from his early efforts and he dominated the second, breaking the Brit twice as he claimed it 6-1.
Evans bounced back in third, however, with some aggressive forehands which kept the Spaniard off balance, he got the break he needed to take the set 6-4.
The 23-year-old was on top in the game he broke Robredo again in fifth game of the fourth set, only for the Spaniard to break back.
Evans showed fight again as he powered his way to another break of serve and this time he held to leave himself one game away from taking the set.
The Brit earned himself two set points, however, nerves got the better of him as hit two unforced errors which allowed the Spaniard to break back.
Robredo sealed the win in clinical fashion taking the fourth set with another break of the 23-year-old to send him through to the fourth round.