Electing to bat in hot and testing conditions, CSK lost early wickets, but skipper Mahendra Singh Dhoni (31 not out) and Bravo, in his 36-ball knock, stitched an unbeaten 91-run fifth wicket partnership to give some respectability to the team total.
Bravo, who brought up first fifty of the season in just 29 balls, smashed 8 fours and a six while Dhoni played the sheet anchors role but was a tad slow in his 37-ball stay.
Earlier Dwayne Smith also struck a 29-ball 40 to give the much-needed boost after CSK were in trouble at 39 for three.
Looking to make full use of the powerplays in the first six overs, Brendon McCullum (12) fell to veteran leg-spinner Pravin Tambe (1-36) after he mistimed a pull to James Faulkner at mid-on.
Suresh Raina (4) was the next to go as Chris Morris (1-19) got the left-hander caught behind in the sixth over.
Young left-arm spinner Ankit Sharma (1-26), who opened the bowling for the Shane Watson-led side, then got Faf du Plessis back in the dug out to reduce CSK to 39 for three.
But just when it looked that Smith, whose innings was laced with 3 fours and 2 sixes, will bail the visitors out of trouble with his big and powerful hitting, RR medium-pacer Faulkner (1-26) got the burly West Indian clean bowled just after the strategic time out post the ninth over.
But Faulkner and company bowled a decent line and length during the course of the 20 overs and South African medium-pacer Morris summed up the team's bowling in the final 6 deliveries, allowing the opposition batsmen only 7 runs.