Curran has played 51 T20 matches taking 53 wickets at an average of 27.16 and an economy of 8.25.
He made his Test debut during the Ashes last December, and has played two Tests, eight One-Day Internationals and six Twenty20 Internationals for England.
Starc, rated as one of the world's finest pacers, was roped in for a whopping sum of Rs 9.4 crore and was tipped to lead the franchise's bowling attack.
"I'm thrilled and excited to be joining Kolkata Knight Riders for the IPL," Curran was quoted as saying by the English county's website.
"I know from chatting to other players I can learn a lot from being in that environment, skills that I hope will benefit Surrey and my chances of representing England again this year and if selected, for next year's World Cup.
"I want to thank Alec Stewart and everyone at Surrey County Cricket Club for supporting my ambitions and allowing me this great opportunity. I can't wait to represent Surrey again when I return, hopefully with a few more tricks up my sleeve."
Stewart, the former England captain who is the director of cricket at Surrey, said: "Losing Tom to the IPL for the first part of the season is not ideal from a Surrey viewpoint but as a club we have a duty to help England and our players in achieving their goals and ambitions.
"The experience he will gain through playing in the IPL can only help develop his game which in turn will give England a greater chance of winning the 2019 World Cup.
"His absence from the Surrey team will allow other players on our staff to stand up, perform and grab their opportunity."
But the 28-year old was ruled out of the fourth Test against South Africa due to a tibial bone stress fracture in his right leg.
KKR take on Virat Kohli-led Royal Challengers Bangalore (RCB) to start their campaign at the Eden Gardens here on Sunday.