Venus dropped serve in the second game, not helped by a double fault at 0-15 to put him under pressure. Tecau's sharpness at the net brought up four break points and while Venus and Klaasen saved three they couldn't save a fourth.
Venus and Klaasen couldn't get a look on Tecau or Rojer's serve in the second set and after Klaasen dropped serve for 0-4 it was effectively over. Venus was broken a second time as he and Klaasen dropped the second set 6-0.
Venus will head home to London for a week of practice ahead of the World Tour Finals at London's 02 Arena. The tournament features the best eight teams in a round robin format with the pair's split into two groups of four.
Last year Venus made his debut in London with American Ryan Harrison on the back of their French Open success, and they won their three round-robin matches before losing in the semi-finals.
It's a lucrative week at the 02 Arena with Venus and Klaasen earning a US$100,000 bonus just for qualifying and US$38,000 for each match win. That increases to $103,000 for winning the semi-final and $200,000 for winning the final with an undefeated champion pair pocketing US $517,000.