But Powell and Axar Patel (24) guided their team towards victory, with the West Indies all-rounder smashing a six to finish off the chase in the 19th over.
Umesh claimed three wickets for KKR and Sunil Narine and Harshit Rana one each.
Batting first, the Knight Riders scored 146 for nine in their 20 overs after their top order struggled to get going.
Kolkata openers Aaron Finch and Venkatesh Iyer returned to the dugout during the power-play.
Then Kuldeep claimed the wickets of Baba Indrajith and Narine (0) with the second and third balls of his spell to reduce them to 35 for four.
Skipper Shreyas Iyer rebuilt the innings with his 42 but fell to Kuldeep in the 13th over, followed by Andre Russell for a duck in the same over.
Nitish Rana launched a blitz at number six, scoring 57 off 34 balls with three fours and four sixes.
He added 62 with Rinku Singh, who scored 23 off 16 balls, to put on a competitive total for the team.
Bangladesh's star left-armer Mustafiz took three wickets in the final over.
"I might have become a better bowler, but I am mentally stronger than before. When you fail in life you pick what you can improve on and that is something I had to improve," said Kuldeep, who was named player of the match.