The Czech was no match for his Canadian opponent - seeded 13th - surrendering four break points as he went down in straight sets on the hard courts at the ATP 500 event.
Pospisil chalked up six aces compared to Berdych's four, and won 82 per cent of his first-service points to advance to the quarter-finals after one hour 21 minutes.
The 24-year-old has now reached the last eight in four of his past five appearances on tour - progressing to the quarters in Atlanta, Bogota, 's-Hertogenbosch and now Washington.
Standing in between Pospisil and the semi-finals is Colombian 10th seed Santiago Giraldo, who eased past Victor Estrella Burgos of the Dominican Republic 6-2 6-0.
Steve Johnson continued his giant-killing run after he edged ninth-seeded Croatian Ivo Karlovic 3-6 7-6 (7-4) 7-6 (9-7).
Unseeded American Johnson stunned counterpart John Isner in the second round and he backed that up with victory over Karlovic in just under two hours on court.
Johnson will face Canadian Milos Raonic in the quarters after the second seed overcame spirited Australian Lleyton Hewitt in straight sets.
Raonic fired down 29 aces as he prevailed 7-6 (7-1) 7-6 (7-3) against the 2004 champion and 14th seed Hewitt.
"I'm happy with a lot of things about the win today, especially the way I was able to fight through the important games," said Raonic.
"I was able to find my rhythm well at the end of each set. He made it difficult for me. The right attitude really got me through today in the key moments."
South African Kevin Anderson and American Donald Young are set to go head-to-head in the next round following their respective wins on Thursday.
Anderson - the seventh seed - disposed of Tunisian Malek Jaziri 6-3 6-4.
That earned a showdown with unseeded Young, who upset Uzbek 15th seed Denis Istomin 6-3 3-6 6-3.
Meanwhile, sixth-seeded Frenchman Richard Gasquet beat American Tim Smyczek 6-3 6-2 to set up a quarter-final meeting with Japan's Kei Nishikori.
Fourth seed Nishikori withstood a stern test from Slovak Lukas Lacko, winning 6-2 2-6 6-3.