The Spaniard looked second best at the beginning of the match, but he dug in to earn a 3-6 7-6 6-3 triumph.
Berdych looked much the stronger of the two players in the opening set, dominating Robredo with his trademark strong forehand to take the set 6-3.
Robredo came out fighting in the second, sealing an early break of serve before cruising into a 5-1 lead.
Berdych started to attack his opponent, and it began to pay dividends with an immediate break of serve, and he broke once more to level the set up at 5-5.
However, the Spaniard went on to take a tie-break to win the set, and he quickly found himself a break up in the decider.
Robredo's variety of shots was proving a problem for Berdych, and he couldn't stop the world number 43 from moving through to a last-eight match with Milos Raonic.