Bopanna and left-handed Nedunchezhiyan were under pressure in the Super tie-breaker but held nerves to pull off a 6-2 3-6 12-10 win over James Cerretani and Philipp Oswald.
After splitting two sets with the giant-killing Austrian-American pair, the Indians were trailing 5-7 when Bopanna hit a stunning backhand winner.
They saved three match points and at 10-10 Bopanna again came up with a backhand winner that allowed Nedunchezhiyan to serve out the match.
The diminutive local boy responded with a big second serve, setting up the finish nicely.
On a day when India's singles challenge ended with the defeat of Yuki Bhambri, another Indian pair made the semifinals.
Divij Sharan and Purav Raja pipped Jurgen Melzer and Renzo Olivo 6-4 7-5 in a tough quarterfinal.
Now two Indian teams in each half of the draw will play semifinals, raising hopes of an all-Indian final for the first time in the history of the tournament.