CSK captain Mahendra Singh Dhoni won the toss and chose to bat under sweltering conditions but his hopes of a strong start were dampened soon.
Kiwi Dasher Brendon McCullum (12) was dismissed in the third over by Royals' veteran spinner Pravin Tambe (1/36).
That was a big blow for CSK but to make matters worse, Indiam middle-order batsman Suresh Raina (4) perished soon, followed by South African Faf Du Plessis (1).
But it was the wicket of West Indian opener Dwayne Smith (40), in the 10th over, that was the biggest setback for CSK.
Royals bowlers were disciplined and kept a tight leash on the CSK batsmen.
The onus was on the fifth-wicket partnership of Dhoni (31 not out) and Dwayne Bravo (unbeaten 62) to lend some sense of respectability to the CSK innings.
The experienced duo were successful in their endeavour, putting up an unbeaten 91-run stand off 65 balls.
They didn't take any particular risk and played themselves in before launching into extravagant strokes in the last three overs.
Bravo was particularly impressive, hitting the formidable Royals attack to all corners of the ground to propel his team them to a competitive total.
Royals' South African pacer Chris Morris was the most impressive giving away a miserly nine runs off his four overs, and claiming Raina's wicket.
Brief scores: CSK 156/4 (D Bravo 62 not out, MS Dhoni 31 not out; Chris Morris 4-0-19-1, James Faulkner 4-0-26-1)