The Czech appeared as through he would make quick work of the world number 42 when he stormed into a 5-0 lead in the first set, but Sousa showed his belligerence.
The 24-year-old slowly settled into the contest by holding for the first time, and then claimed a break back to take the scores to 5-2, but Berdych ensured that his comeback would end prematurely.
In an epic eighth game of the set, which went to deuce on five occasions, the former Wimbledon finalist snatched a break point at the third time of asking to move ahead in the match after 38 minutes.
Sousa continued his resilient response by going blow for blow with Berdych from the back of the court as they exchanged games early in the second, until the world number seven broke for a fourth time.
Although Sousa again did not roll over and broke back immediately, Berdych upped the ante to overpower the Portuguese and thump his way to a quarter-final meeting with American John Isner.