Chahar struck very early by getting rid of Rahane.
Jos Buttler made an electric start but was dismissed for 23.
CSK kept striking at regular intervals to take control of the contest.
Jadeja bowled a top spell, picking up his 100th IPL wicket in the process.
Shreyas Gopal's little cameo in the end helped RR post 151.
Shane Watson was dismissed for a duck in the first over of the chase.
Royals also got rid of Raina and Faf du Plessis early to reduce CSK to 15/3.
Stokes took a blinder to get rid of Jadhav to pile on the misery as CSK were reduced to 24/4.
MS Dhoni then rescued CSK with a crucial half-century.
Rayudu chipped in with his maiden fifty of the season as well to put CSK back on track.
Even though the half-centurions departed, Santner and Jadeja completed the job in the final over with the New Zealand allrounder hitting a six off the final ball to seal the win.