Bopanna and Qureshi, seeded fifth, defeated Australian Paul Hanley and his Belgian partner Dick Norman 6-2 6-3 in a contest lasting a little over an hour.
They will now face the unseeded British pair of Colin Fleming and Ross Hutchins, who got a walkover from their 13th seeded Spanish rivals Marcel Granollers and Marc Lopez.
Bopanna and Qureshi converted all three break points that came their way while holding their own serve all through the match.
The fifth seeds struck 24 winners against their opponents' 10. However, there was disappointment for India in the women's doubles event as Sania and her Russian partner Elena Vesnina lost to the Czech pair of Iveta Benesova and Barbora Zahlavova Strycova.
The sixth seeded Indo-Russian team went down 6-7 (4) 6-7 (5) against their ninth seeded rivals. Sania and Vesnina managed to convert just one break point out of the nine they earned and committed as many as 31 unforced errors during the match.