Sania and Bhupathi lost 3-6, 6-7 (5) to the Czech pair of Lucie Hradecka and Frantisek Cermak in a contest that lasted a little over an hour.
However, Somdev, who lost in the men's singles opening round against Andy Murray, ensured that India didn't go winless on day four of the season's last Grand Slam.
Somdev and his Filipino partner, Treat Conrad Huey, defeated Russians Igor Andreev and Igor Kunitsyn 6-3 6-4 to enter the men's doubles second round.
The Indo-Filipino pair broke their rivals four times in the course of the match that lasted one hour and seven minutes.
Somdev and Huey will face 14th seeds Argentines Juan Ignacio Chela and Eduardo Schwank in their next match.