Rahane stood out as the day's best performer with his 27-ball 61 on debut for Chennai, guiding them to their target of 158 with 11 balls to spare at Mumbai's Wankhede Stadium.
"Me and Jinks (Ajinkya) spoke at the start of the season and I told him to play to his strengths, use your ability to manipulate the fields," said CSK captain MS Dhoni.
"I told him to go and enjoy, don't take stress and we'll back you."
Coming in after the fall of Devon Conway for a duck, Rahane tore into the Mumbai attack, which was without Jofra Archer after the England pace bowler was rested due to discomfort in his elbow.
Mumbai skipper Rohit Sharma said Archer, who returned to the team in the previous match after missing last season, not playing was "just a precaution".
Rohit said the five-time champions need to be more "brave" in their upcoming matches.
Chennai's England stars Ben Stokes and Moeen Ali missed out due to injuries and South Africans Dwaine Pretorius and Sisanda Magala made the team.
Delhi 'long way off'
Ravindra Jadeja made an impression with his left-arm spin as he took the key wickets of Ishan Kishan (32), Cameron Green, out for 12 after an excellent caught and bowled, and Tilak Varma (22) to keep Mumbai to 157-8.
Jadeja, who was named man of the match, was ably supported by New Zealand left-arm spinner Mitchell Santner, who returned figures of 2-28.
Tim David's late cameo of 31 boosted Mumbai's total but was still not enough to challenge the Chennai batting.
Rahane, who was signed by Chennai for his base price of $61,094 in the December auction, smashed seven fours and three sixes.
Opener Ruturaj Gaikwad made an unbeaten 40 to move to the top of the batting charts with 189 runs this season as he jumped over Buttler (152) and Warner (158).
In the first match of the day in Guwahati, last year's runners-up Rajasthan posted 199-4 after an attacking 98-run opening stand between the left-handed Jaiswal (60) and Buttler (79).
Jaiswal struck five boundaries in the first over and Buttler soon took charge and, despite a middle-order stutter, took on the bowlers before Shimron Hetmyer hit an unbeaten 39.
New Zealand's left-arm quick Trent Boult led the bowling charge with a double-wicket maiden to help restrict Delhi to 142-9 as Rajasthan bounced back from their previous loss.
"We are a long way off right now," Delhi coach Ricky Ponting said.
"I can't put my finger on why? We need to do a bit more soul searching as a group.
"We have to think about the players we are putting on the ground because what we have done so far hasn't worked."
Rajasthan, who won the first edition of the IPL in 2008 under late Australian great Shane Warne, have won two of their three matches.