The South African struck his second six, against medium pacer Siddharth Trivedi, before Kevon Cooper brought to an end his stay by having the batsman caught by Goswami.
Cooper then sent back Raina, thanks to Ajinkya Rahane's excellent catch in the deep, but skipper Mahendra Singh Dhoni and Ravindra Jadeja denied the visitors any further success while taking Chennai home.
Earlier, the Super Kings rode on a disciplined bowling display to restrict Rajasthan to a modest total despite a half century by Owais Shah on a swealtering day.
Electing to bat on what looked like a slow pitch, the Rajasthan Royals batsmen started off in brilliant fashion, scoring 15 runs off Dwayne Bravo's first over.
They were 34 for no loss by the fourth over before Ravichandran Ashwin put the brakes on the scoring rate.
Ashwin gave his team a vital breakthrough when he dismissed an in-form Rahane (15) off the third ball of his spell.
Image: Rajasthan Royals cricketer Ajinkya Rahane (R) is congratulated by his teammates after he dismissed Chennai Super Kings batsman Suresh Raina during the IPL Twenty20 cricket match between Chennai Super Kings and Rajasthan Royals at The M.A.Chidambaram Stadium in Chennai on April 21, 2012.