Klaasen improved on his previous best ranking of 20th, after reaching the quarterfinals of the Paris Masters with American partner Eric Butorac over the weekend.
They failed to progress to the next round after they were beaten 7-5, 1-6, 6-10 by the doubles pairing of Jean-Julien Rojer and Horia Tecau.
Last month they lifted their second ATP World Tour title of the year at the Stockholm Open, after defeating Treat Huey and Jack Sock 6-4, 6-3 in the final.
Meanwhile, South Africa’s top singles player Kevin Anderson moved back to his best ranking of 16th after reaching the quarterfinals of the Paris Masters, where he lost to Tomas Berdych of the Czech Republic.
Berdych defeated Anderson 6-7 (4), 6-4, 6-4, recording his 12th victory over Anderson in as many meetings.