The number six seed took one hour and 41 minutes to set up a clash with Milos Raonic in the next round.
There was little to separate the two players in the opening set, with both giving away a service game in the early stages. However, both Berdych and Bautista Agut were dominant on their serves thereafter as the set went to a tie-break, which Berdych won 9-7.
Berdych made the brighter start to the second set as he broke Bautista Agut's serve in the opening game. The Czech looked comfortable at 2-0 up, but he was pegged back by his opponent, who broke back in the fourth game.
The remainder of the set looked on course to follow the same pattern as the first, with both players holding serve throughout. However, unforced errors from Bautista Agut in the ninth game allowed Berdych to make a crucial break and move 5-4 ahead.
Berdych did not let his advantage slip, serving out the set with his first match point to move into the last eight.