The Paes-Klaasen combination came from a set down to beat Bhupathi and Myneni 1-6 6-1 10-7 in the keenly-contested quarterfinal duel.
The Indo-South African pair will next face unseeded Spanish duo of Pablo Carreno Busta and Guillermo Garcia-Lopez in the semifinals.
Busta and Garcia-Lopez had defeated Ricardas Berankis of Lithuania and Mate Pavic of Croatia 6-4 0-6 10-4 in another quarterfinal match.
But Paes used all his experience and combined well with Klaasen to forge a comeback in the match.
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Paes-Klaasen broke Bhupathi and Myneni twice to pocket the second set and take the match into super tie-breaker.
In the super tie-breaker, Paes and Klaasen kept their composure to seal the tie in their favour.
Meanwhile in another doubles quarterfinal match earlier in the day, Indo-Canadian pair of Purav Raja and Adil Shamasdin defeated Spaniard Roberto Bautista Agut and Stanislaus Wawrinka of Switzerland 7-6 4-6 10-4 to book a place in the last four round.