Kohli led a bold run chase on Saturday's gripping final day of the opening Adelaide Test, but the tourists ultimately went down by 48 runs to the home side chasing a formidable 364 for victory.
Kohli's highest Test score, a majestic 141, raised hopes of a come-from-behind Indian win, becoming just the fourth Indian to score centuries in each innings of a Test.
"If we were able to pull this off, today would have been one of the most special moments in my life," Kohli told reporters after the loss.
"But I'm really proud of the way the boys played."
At no stage did the tourists consider batting for a draw, despite being set a daunting target to win on the last day on a wearing pitch.
No team has scored more than 315 runs and won in Adelaide since 1902, but Kohli almost singlehandedly brought off an astonishing win with his second century of the match before he was caught on the boundary rope in the 82nd over.
"At no point did we back off. And we are not going to back off either," Kohli said.
"If you see cricket like this for the rest of the series, it's in the balance. The guys are believing that.
"And unless you believe in something, there's no chance of achieving it.
"If at any stage that we had in mind that we might play for a draw, we would have lost by 150 runs, to be honest.
"So we went for it. That is what I am happy about."
Dhoni, who missed the series opener because of a fractured thumb, will return to the captaincy for the second Test, starting in Brisbane on Wednesday.
"I would be very happy that he is going to be fit and available for the second Test," Kohli said.
"It doesn't matter if I have the 'c' in front of my name or not.
"That doesn't change anything as far as my mindset is concerned.
"I have enough friendship and enough respect between the team that I can go and speak to them every now and then and keep them on the same page.
"That is my responsibility, that is something I will look to do even if I'm not captain."
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