Former Australia captain Smith and his deputy David Warner were handed 12-month bans by Cricket Australia for their roles in a ball-tampering scandal in South Africa.
The pair have also been banned from this year's IPL starting on Saturday with a match between defending champions Mumbai Indians and twice winners Chennai Super Kings.
Klaasen was bought for a modest 5 million Indian rupees (£54,654) to replace Smith whom Rajasthan had retained for 120 million rupees to lead the side in the eight-team tournament. India middle-order batsman Ajinkya Rahane will captain the side.
Klaasen, 26, has played four one-day and three Twenty20 Internationals for South Africa, all against the visiting Indian side last month.
Warner, who quit as Sunrisers Hyderabad captain following the scandal in South Africa, has been replaced by England opener Alex Hales in the IPL side.
(Reporting by Amlan Chakraborty in New Delhi; editing by Ed Osmond)
