Both Prajnesh Gunneswaran and Ramkumar had lost their singles matches on Friday and to stay alive in the tie, the Indians needed to win the doubles.
With this defeat, India's challenge has ended at the last Grand Slam of the season.
The 14th seed Indo-Croatian pair won 6-3 4-6 6-4 over their unseeded rivals in one hour and 56 minutes on Saturday night.
Aditi Ashok to Abhinav Bindra: Heartbreaking instances when Indian athletes finished 4th at Olympics
There is this popular idiom - So near, yet so far. That's something many Indian athletes over the years have learnt the hard way, especially at showpiece events like the Olympics.
Regardless of who messed up, India never had chance to send men's doubles team to Tokyo Olympics: Somdev Devvarman
Somdev Devvarman rallied behind Rohan Bopanna and said there was no need for the AITA to give false hopes of qualification.
For over 50 years tennis players have suffered because of AITA's incompetence, alleges Rohan Bopanna
Rohan Bopanna slammed the AITA for playing with the careers of players, and he went on to label the federation incompetent.
AITA says why did Bopanna not qualify on his own after player accuses it of misleading on nomination
The All India Tennis Association (AITA) had paired Bopanna and Divij Sharan for the men's doubles event but the team failed to make the cut because of its low combined rank of 113.
Bopanna said International Tennis Federation never accepted an entry for him and Sumit Nagal and had made it clear that no changes were allowed after the deadline of nomination.
Bopanna had said ITF never accepted an entry for him and Sumit Nagal and had made it clear that no changes were allowed after the deadline of nomination.
Rohan Bopanna and Divij Sharan, with a combined rank of 113, till Friday morning formed the fourth alternate team.
The contest resumed with the 14th seed pair of Jean Julien-Roger and Andreja Klepac having a one-set lead and they finished on the right side of the result with a 6-3 3-6 11-9 scoreline.
Ankita Raina said it was a privilege to play against the Indian mixed doubles pair of Sania Mirza and Rohan Bopanna.
Bopanna was clearly the best player on the court with his powerful serve and solid ground strokes from the baseline as well as the ability to execute a superior net game.
Indian faceoff at Wimbledon: Sania Mirza, Rohan Bopanna to lock horns with Ankita Raina, Ramkumar Ramanathan in first round
Earlier in the day, Sania and her doubles partner Mattek-Sands sailed into the second round of the ongoing Wimbledon Championships.
Meanwhile, Sania Mirza and her American partner Bethanie Mattek-Sands stunned sixth seeds Desirae Krawczyk and Alexa Guarachi to make the second round of the women's doubles event.