This was India's sixth Test win on Australian soil, coming after some tense moments thanks to the Australian lower order's resistance.
"These things happen in Tests, ups and downs through a game. Odds were stacked against them. They fought really well but we executed our plans well," Kohli said at the post-match presentation ceremony.
Asked if he felt the heat, the India skipper said, "I wouldn't say I was cool as ice but you just don't try to show it. Jasprit was getting worked up in his last over but I just told him to relax. Super proud (of the bowlers), to have four bowlers and take 20 wickets is a great achievement. Something that we haven't done in the past.
Kohli said India were "collectively better" in the Test and hoped his batsmen would step up regularly after the bowlers played a part.
"(It) shows us that if the batsmen step up regularly, we will be gunning for a win in every Test match. Collectively we were the better team and deserved to win," Kohli said.
It was Cheteshwar Pujara's century, which bailed India out of a difficult situation on the first day, and the senior pro contributed handsomely in the second essay too, scoring 71 and attracting Kohli's praise.
"It was priceless from Pujara. His grit and determination brought us back in it. We always knew that runs on the board would make the home side tentative. Any lead was gold and we got 15.
"In the second innings again, he and Rahane batted well."
Did he think the target of 323 was enough and Kohli replied, "I think our lower middle order and lower order could have done better. We could have added another 30-35 runs more which could have taken the game totally beyond Australia.
"So these are things that we have to think about going to Perth but if someone had told me before the series that I would be 1-0 up as soon we started the series, I would have taken it with both hands."
Australian captain Tim Paine was a disappointed man, his team having come close to an improbable win.
"It's pretty shattering. But India thoroughly deserved to win. We thought we could do it. But our batters couldn't bat for long enough with our tail to win. I thought Pujara was probably the difference between the two sides.
"We go to Perth with a belief we can still win this Test series."
Man of the Match Pujara said past experience of playing in this part of the world helped.
"I think playing here before has helped me a lot. I think what has helped me is the preparation. Ultimately (for) winning the Test match credit (goes) to all the bowlers.
"The first innings lead of 15 gave us the belief batting. I always back my ability and I've got enough experience," Pujara said.
His father played an influential in his growth as a cricketer, and Pujara thanked him.
"It means a lot, he'll be a proud father. I'd like to thank him for all his support throughout my career.