Thakur hit nine fours and three sixes during his knock of 68, powering a recovery from a worrisome 89-5 in the 12th over.
Opener Rahmanullah Gurbaz propped up the innings with 57 off 58 balls but the Afghan's dismissal, along with Andre Russell for a duck the next ball, left Kolkata looking in trouble.
Thakur's dominant show put the hosts back in control and a sixth-wicket partnership with Rinku Singh notched 103 runs off only 47 balls.
"We spoke about taking the innings to the end and not giving away more wickets," Singh said.
"My plan was to just stick around as he was striking the ball well. So I kept giving him the strike."
Bangalore crashed to 123 all out in reply as the home team's spinners ran amok, with Varun Chakaravarthy picking 4-15 in less than four overs.
See the full scorecard on Flashscore.
Kolkata's teenage leg spinner Suyash Sharma took 3-30 on his IPL debut and West Indies veteran Sunil Narine went 2-16.
Virat Kohli (21) and Faf du Plessis (23) gave Bangalore a quick start, adding 44 off 29 balls before the bowling attack found its groove when Narine bowled the former Indian captain.
Chakaravarthy dealt du Plessis the same fate three balls later and returned in the eighth over to skittle out Glenn Maxwell (5) and Harshal Patel (0).
Kolkata next take on defending champions Gujarat Titans in Ahmedabad on Sunday and will have English opener Jason Roy available.
He was signed as a replacement for Bangladesh all-rounder Shakib Al Hasan, who pulled out of the tournament earlier this week owing to his international commitments with Bangladesh.
