The fifth seed Indian duo, which got a bye in the first round, beat the Cyprus-Scot combo 6-3 7-6(5) in one hour and 12 minutes in the pre-quarterfinal round.
The second set was stretched to tie-breaker as Murray and Baghdatis fought well to save five break points that Bhupathi and Bopanna created.
Leander Paes is also in the fray along with a new partner in South Africa's Kevin Anderson.
They are unseeded and face French pair of Julien Benneteau and Nicolas Mahut.