The Paes-Klaasen combination came from a set down to beat Bhupathi and Myneni 1-6, 6-1, 10-7 in the keenly-contested quarter-final duel.
The Indo-South African pair will next face unseeded Spanish duo of Pablo Carreno Busta and Guillermo Garcia-Lopez in the semi-finals.
Busta and Garcia-Lopez had defeated Ricardas Berankis of Lithuania and Mate Pavic of Croatia 6-4, 0-6, 10-4 in another quarter-final match.
Paes and Klaasen were slow to get off the blocks as Bhupathi and Myneni broke their rivals twice out of four opportunities to pocket the first set 6-1.
But Paes used all his experience and combined well with Klaasen to forge a comeback in the match.
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 quarter-final 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.
After losing the match, Bhupathi said Paes' vast experience went against them.
"We know each others strengths and weakness and perhaps this is the best doubles match that was played so far in the tournament history for 20 years," Bhupathi said.
"He (Paes) is known for playing and winning the big points. We lost today manly because of that," he added.
Image: Leander Paes and Raven Klaasen celebrate after winning the game against Mahesh Bhupathi and Saketh Myneni during the Chennai Open
Photograph: PTI
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