The British pair took the first set of the clash on a tiebreak, but lost the second in similar fashion as all 24 games followed serve.
Klaasen and Venus were seeded third in the tournament and showed why in the deciding tiebreak, breaking the Murrays’ serve to open up a 3-0 lead.
Undeterred, the brothers bounced back to take a 7-5 lead but the South African and New Zealander won five points in a row to prevail 6-7 (3) 7-6 (6) 10-7.
Meanwhile, Kyle Edmund also tumbled out in Washington at the hands of German lucky loser Peter Gojowczyk.
The 24-year-old’s disappointing summer continued with a three-set defeat in the quarter-finals, going down 6-3 4-6 6-3 to his opponent, ranked 122 in the world.
Gojowczyk wrapped up the first set in 32 minutes but Edmund drew the match level.
A solitary break in the third gave the 30-year-old the upper hand, as Edmund went on to save four match points but netted the fifth.
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