Excitement was high in India as the tournament proper kicked off on Wednesday with a meeting of two home sides.
And the game was left in the balance when MS Dhoni, who lost the toss, struck a fine 35 not to secure a total of 157-4 for Chennai.
But an attainable victory target suddenly seemed unlikely for the Knight Riders as Ashish Nehra grabbed three quick wickets.
Ten Doeschate, though, kept his head amid the flurry of falling wickets, striking a composed 41-ball 51 not out.
And with Andre Russell providing the fireworks - clearing the rope five times in a stunning 58 - Kolkata won by three wickets.
Every batsman who walked to the crease for Chennai made a contribution of some sort, with Francois du Plessis' 14 the poorest effort.
But it was Dhoni, building on the platform laid by Dwayne Smith (20), Brendon McCullum (22) and Suresh Raina (28), who made the most decisive contribution.
The wicketkeeper-batsman struck three fours and two sixes - including a huge one into the upper tier from the innings' final ball.
Dwayne Bravo did his bit, too, clearing the rope twice in a run-a-ball unbeaten 28.
Any hopes Kolkata had were immediately undermined as Nehra (4-21) began to make his mark.
Gautam Gambhir was his first victim, hitting the paceman straight to Bravo at mid-on.
Manish Pandey, whose superb 94 in the final earned the Knight Riders their IPL crown, fell in identical fashion next ball for a golden duck.
Ishant Sharma got in on the act in the next over, Yusuf Pathan edging to slip, before Nehra had a third man caught at mid-on, Manvinder Bisla departing.
But after Suryakumar Yadav fell for 19, Russell launched his stunning intervention alongside the more reserved Ten Doeschate.
The West Indies all-rounder could not see it through to the end, bowled by the returning Nehra, but Ten Doeschate was on hand to finish the job.