Kohli has hit 6283 runs in 207 IPL matches since he joined Bangalore in the inaugural edition in 2008 and led them to a runners-up finish in 2016.
"It's a great time to regroup and restructure the franchise for the next three years," said Kohli.
"I will definitely (play for RCB). For me loyalty matters and my commitment is with this franchise till the last day I play IPL."
Narine returned figures of 4-21 as Kolkata kept down Bangalore to 138-7 and then hit 26 off 15 balls in the team's chase that was completed with two balls to spare.
Bangladesh all-rounder Shakib Al Hasan hit the winning run with skipper Eoin Morgan at the other end as Kolkata booked a meeting with Delhi Capitals in the second qualifier on Wednesday.
Narine stood out with his off spin as he struck in each of his four overs to send back Srikar Bharat (9), Kohli (39), AB de Villiers (11) and Glenn Maxwell (15).
Narine bowled Kohli with flight and spin as the star batsman walked off for one final time as captain of Bangalore since taking over the role in 2013.
He then bowled South African veteran De Villiers and had the in-form Maxwell caught at short third man.
In reply, openers Shubman Gill (29) and Venkatesh Iyer (26) gave Kolkata a strong start adding 41 before the first wicket fell in the sixth over.
The innings wobbled slightly before Narine, coming in at number five, hit three sixes off Dan Christian to ease the pressure.
He was bowled by Mohammed Siraj who also sent back Dinesh Karthik in the same over but Morgan and Shakib kept their cool to take the two-time IPL champions home.
Kolkata have won five of their seven league games in the UAE leg to make the playoffs.
"The cricket we have played has surprised everybody," said England's white ball captain Morgan.
MS Dhoni's Chennai Super Kings have made the final on 15 October after they beat Delhi in the first playoff on Sunday.