Kohli (67) in company of AB de Villiers (59 not out) scripted a superb run chase, sharing a 85-run stand to lay the foundation, while Marcus Stoinis (28 off 16) played a crucial cameo as RCB overhauled the target of 174 with four balls to spare and eight wickets in hand.
"It is a great feeling to get across the line. Have been unlucky in couple of games. Won't say we've been unlucky in every game, but a couple of games we should've closed out before this," Kohli said at the post-match presentation on Saturday.
"After so many setbacks, the guys had desire. That's one word we spoke about. We thought 190 will be par and to restrict them to 170 was a great effort. To pick up 4 wickets in 8 overs in the middle for 60 runs was a great effort."
AB de Villiers, who was adjudged the Man of the match, said it was a long wait for the team.
"Waited for a long time for this, so we're very happy. It's a very small step in the right direction, but also a big one," he said.
"I've got too much respect of the game to be upset about that, to get ahead of myself, to get too hard on myself. It needs one knock to get back in that confidence zone, and hopefully I can maintain that now."
De Villiers had retired from international cricket last year to prolong his career in T20 cricket.
"I've decided not to play international cricket - to be as sharp as I can in the other forms of cricket around the world. Can't keep playing 10-11 months a year and keep being sharp after 15 years of my career."
RCB received a big boost when the seasoned South African pacer Dale Steyn was named as a replacement for injured Australian Nathan Coulter-Nile.
"It was a fantastic move to get Dale involved. We all know what he's capable of. One thing for sure is he'll give 200% with every ball. We know what we have to do.
"We have to win a lot of games of cricket. The next step is Mumbai, the Wankhede stadium. We love playing cricket there," said de Villiers.
"We let ourselves down in the field. We dropped catches and let a lot of balls go through us. The dew helped the wicket on. It was really dry in the first half. It gets better to bat in the second half," he said.
Asked why Sarfraz was chosen to bowl the final over, Ashwin said: "There was only 6 runs to go and the ball was sliding on nicely. I thought the the best way against AB and Stoinis was to try and go with a leggie, not that Sam can't defend.
"We tried with wrist spinners and even I came to bowl, but they were smart enough to knock around and not take any chances.
"With 38 needed off 3 overs, I had to back my best bowlers to do the job, didn't quite work out today."
Talking about the coming games, Ashwin said: "We need to string some wins. It's not a bad time to get some momentum back."
RCB teammate Stoinis praised de Villiers for his unbeaten knock.
"I said to him (AB), that he is probably the best player of our generation and he showed it today," he said.
On RCB finally getting a win, he added: "Yes, I was telling Gary (Kirsten) that I had brought two points for the side after my recent trip and we got it today (chuckles).
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)