The seventh-seeded Indian combination got the better of Marcel Felder of Uruguay and Tunisia's Malek Jaziri 6-0 7-6 (1) 6-2 in an hour and 36 minutes at the All England Club here.
The duo will take on the winner of the match between Russia's Mikhail Elgin and Uzbekistan's Denis Istomin and Spaniards Pablo Andujar and Guillermo Garcia-Lopezin in the second round.
Bhupathi and Bopanna converted five of the nine break points they got, while their rivals failed to convert any of the five opportunities that came their way.
The Indians also won as many as 101 points as opposed to just 66 won by their unseeded opponents.
Dominating from the word go, the Indian duo took just 17 minutes to wrap up the first set after converting three of the four breakpoint chances.
However, Felder and Jaziri managed to take the second set to the tie-breaker, even as Bhupathi and Bopanna double faulted thrice, besides making three unforced errors.
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The Indians, however, managed to keep their composure and sealed the second set in the tie-breaker to go 2-0 up.
Determined not to let the advantage slip out of their hands, Bhupathi and Bopanna hardly gave any breakpoint chance to Felder and Jaziri in the third set, while converted two of the five chances they got to seal the set and the match in their favour.