Kohli returned to form after back-to-back ducks as he steered his team home in a chase of 193 in Raipur with an excellent 60-ball innings.
The 37-year-old scored his ninth IPL hundred, hitting 11 fours and three sixes.
Kohli also became the fastest player to reach 14,000 runs in T20 cricket as he achieved the feat in 409 innings, bettering Chris Gayle's 423.
"I just love batting," said Kohli. "What an honour to be playing at this level. This is all I have done all my life. I want to make the most of every day."
Bengaluru notched their eighth victory in 12 matches this season to usurp Gujarat Titans at the top of the 10-team table on net run-rate, moving closer to securing a playoff spot.
Three-time champions Kolkata suffered their sixth defeat in 11 matches as their slim chances of making the playoffs suffered a blow.
The top four will make the playoffs, with the final to be played in Ahmedabad on May 31st.
Kohli got off the mark with a single in the first over of the chase, punching the air in mock celebration after his successive ducks.
"There's a reason why people say pressure is a privilege," said Kohli.
"Good pressure always makes you improve your game. You go a level up, you keep improving your game. I was nervous, I just wanted to get off the mark."
He lost his opening partner, Jacob Bethell, for 15 but put on 92 runs with Devdutt Padikkal, who made 39, for the second wicket.
Kohli was dropped on 21 and made Kolkata pay, bringing up his fifty in 32 balls.
Bengaluru lost Padikkal, skipper Ruturaj Gaikwad and then Tim David to a stunning one-handed diving catch at backward point by Manish Pandey.
But Kohli kept calm to reach his first hundred of the season, soaking in the applause before wicketkeeper-batter Jitesh Sharma hitting the winning runs.
Earlier, Kolkata posted 192-4 after a 46-ball 71 by Angkrish Raghuvanshi.
Raghuvanshi put on strong partnerships, including with Cameron Green, who scored 32, and Rinku Singh, who finished with a flourish in his unbeaten, 29-ball 49.
Pace bowlers Bhuvneshwar Kumar, Josh Hazlewood and Rasikh Salam took one wicket each.